Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

§1: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 The Quantized Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Schur Algebras and q-Schur Algebras . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Coxeter Systems and their Properties . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

§2: A New Setting for the q-Schur Algebra . . . . . . . . . . . . . . . . . 24

2.1 The classical Schur algebra as a subspace of tensor matrix space . . . . . . . . . . 24 2.2 The v-Schur algebra as a subspace of tensor matrix space . . . . . . . . . . . . . 28 2.3 On certain subalgebras of the q-Schur algebra . . . . . . . . . . . . . . . . . . . 44

§3: A Straightening Formula for Quantized Codeterminants . . . . . . . . . 47

3.0 Conventions and Review of §2 . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Hecke algebras and codeterminants of dominant shape . . . . . . . . . . . . . . 48 3.2 The Quantized Straightening Formula . . . . . . . . . . . . . . . . . . . . . . 55

§4: q-Schur Algebras as Quotients of Quantized Enveloping Algebras . . . . . 63

4.0 Conventions and Review of §3 . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1 The restriction of θ to U^{−}, U^{0} and U^{+} . . . . . . . . . . . . . . . . . . . . . 64 4.2 Codeterminants, explicit surjectivity and ker (θ) . . . . . . . . . . . . . . . . . 73 §5: q-Weyl modules and q-codeterminants . . . . . . . . . . . . . . . . . 78

5.1 Quantized Left Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Quantized Right Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 q-Codeterminants and q-Weyl modules . . . . . . . . . . . . . . . . . . . . . . 90 5.4 Remarks on the case n < r . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

§6: Cellular inverse limits of q-Schur algebras . . . . . . . . . . . . . . . 96

6.1 The strong straightening result . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 Cellular algebras and quasi-hereditary algebras . . . . . . . . . . . . . . . . . . 99 6.3 Epimorphisms between v-Schur algebras . . . . . . . . . . . . . . . . . . . . 105 6.4 Cellular inverse limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Acknowledgements

Above all, I would like to thank my supervisor, Professor R.W. Carter, for introducing me to such an interesting area, for teaching me to write good mathematics, and for making so many helpful comments and corrections to earlier drafts of this thesis.

I also thank the Mathematics Institute for providing an excellent working atmosphere, without which I would not have been able to make so much progress so quickly. I would like to thank all my friends and colleagues there, especially John Cockerton and Robert Marsh, for many interesting discussions.

I am grateful to Professor S. Donkin, Professor R.C. King and Professor G.I. Lehrer for their interest in my research and for their helpful and encouraging comments during the preparation of this thesis.

Finally, I would like to thank the Engineering and Physical Sciences Research Council for funding my research.

Declaration

The material in Chapter 1 is expository. The material in Chapters 2, 3, 4, 5 and 6 is to the best of my knowledge original, except where otherwise indicated.

Summary

The main aim of this thesis is to investigate the relationship between the quantized enveloping algebra U(gl_{n}) (corresponding to the Lie algebra gl_{n}) and the q-Schur algebra, S_{q}(n, r). It was shown in [BLM] that there is a surjective algebra homomorphism

θ_{r} : U(gl_{n}) → ℤ[v, v^{−1}] ⊗ S_{q}(n, r),

where q = v^{2}.

§1 is devoted to background material.

In §2, we show explicitly how to embed the q-Schur algebra into the r-th tensor power of a suitable n × n matrix ring. This gives a product rule for the q-Schur algebra with similar properties to Schur’s product rule for the unquantized Schur algebra. A corollary of this is that we can describe, in §2.3, a certain family of subalgebras of the q-Schur algebra.

In §3, we use the product rule of §2 to prove a q-analogue of Woodcock’s straightening formula for codeterminants. This gives a basis of “standard quantized codeterminants” for S_{q}(n, r) which is heavily used in chapters 4, 5 and 6.

In §4, we use the theory of quantized codeterminants developed in §3 to describe preimages under the homomorphism θ_{r} and the kernel of θ_{r}.

In §5, we use the results of §3 and §4 to link the representation theories of U(gl_{n}) and S_{q}(n, r). We also obtain a simplified proof of Dipper and James’ “semistandard basis theorem” for q-Weyl modules of q-Schur algebras.

In §6, we show how to make the set of q-Schur algebras S_{q}(n, r) (for a fixed n) into an inverse system. We prove that the resulting inverse limit, Ŝ_{v}(n), is a cellular algebra which is closely related to the quantized enveloping algebra U(sl_{n}) and Lusztig’s algebra U̇.

1. Preliminaries

In this chapter, we introduce the necessary background material for the rest of this thesis. Most of this material is expository. In §1.1, we introduce the quantized enveloping algebra, and in §1.2, we introduce the Schur algebras and q-Schur algebras. In §1.3, we state some well known properties of Coxeter systems, which we use to analyse symmetric groups and their associated Hecke algebras. Finally, §1.4 is devoted to combinatoric definitions.

The symbols q and v always represent indeterminates unless stated otherwise, and are related via v^{2} = q. Throughout the thesis, we denote ℤ[v, v^{−1}], the ring of Laurent polynomials over ℤ, by A.

§1.1 The Quantized Enveloping Algebra

1.1.1 Definition of the Quantized Enveloping Algebra We now define the algebra U(gl_{n}) over ℚ(v) as in [D4]. It is generated by elements

E_{i}, F_{i}, K_{j}, K_{j}^{−1},

(where 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n) subject to the following relations:

K_{i}K_{j} = K_{j}K_{i},  (1)
K_{i}K_{i}^{−1} = 1,  (2)
K_{i}E_{j} = v^{ϵ^{+}(i,j)}E_{j}K_{i},  (3)
K_{i}F_{j} = v^{ϵ^{−}(i,j)}F_{j}K_{i},  (4)
E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K_{i}K_{i+1}^{−1} − K_{i}^{−1}K_{i+1}}{v − v^{−1}} ,  (5)
E_{i}E_{j} = E_{j}E_{i} if |i − j| > 1,  (6)
F_{i}F_{j} = F_{j}F_{i} if |i − j| > 1,  (7)
E_{i}^{2}E_{j} − (v + v^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 if |i − j| = 1,  (8)
F_{j}^{2}F_{i} − (v + v^{−1})F_{j}F_{i}F_{j} + F_{i}F_{j}^{2} = 0 if |i − j| = 1.  (9)

Here,

ϵ^{+}(i, j) := \{ \begin{matrix}1 & if j = i; \\ −1 & if j = i − 1; \\ 0 & otherwise; \end{matrix}

and

ϵ^{−}(i, j) := \{ \begin{matrix}1 & if j = i − 1; \\ −1 & if j = i; \\ 0 & otherwise. \end{matrix}

The algebra is also equipped with two coassociative comultiplications. One, Δ : U → U ⊗ U, has the following effect on the generators:

\begin{aligned}Δ(E_{i}) & = 1 ⊗ E_{i} + E_{i} ⊗ K_{i}K_{i+1}^{−1}, \\ Δ(F_{i}) & = K_{i}^{−1}K_{i+1} ⊗ F_{i} + F_{i} ⊗ 1, \\ Δ(K_{i}) & = K_{i} ⊗ K_{i}, \\ Δ(K_{i}^{−1}) & = K_{i}^{−1} ⊗ K_{i}^{−1}. \end{aligned}

The other map, Δ′ : U → U ⊗ U, is defined by

\begin{aligned}Δ′(E_{i}) & = K_{i}K_{i+1}^{−1} ⊗ E_{i} + E_{i} ⊗ 1, \\ Δ′(F_{i}) & = 1 ⊗ F_{i} + F_{i} ⊗ K_{i}^{−1}K_{i+1}, \\ Δ′(K_{i}) & = K_{i} ⊗ K_{i}, \\ Δ′(K_{i}^{−1}) & = K_{i}^{−1} ⊗ K_{i}^{−1}. \end{aligned}

We will not be using Δ′ in this thesis, but we include it for completeness because many authors prefer it to Δ.

The maps Δ and Δ′ are also algebra homomorphisms. This means that if M is a U-module, we can make M^{⊗r} into a U-module via

u.(m_{1} ⊗ ⋯ ⊗ m_{r}) = Δ^{r−1}(u)(m_{1} ⊗ ⋯ ⊗ m_{r}).

This is well-defined because Δ is coassociative. Note that Δ^{r−1} is an algebra homomorphism U → U^{⊗r}. It may be checked by a simple inductive argument that the effect of Δ^{r−1} on the generators E_{i}, F_{i}, K_{i}, K_{i}^{−1} is as follows:

Δ^{r−1}(E_{i}) =(1 ⊗ ⋯ ⊗ 1 ⊗ E_{i}) + (1 ⊗ ⋯ ⊗ 1 ⊗ E_{i} ⊗ K_{i}K_{i+1}^{−1}) + ⋯
⋯ + (E_{i} ⊗ K_{i}K_{i+1}^{−1} ⊗ ⋯ ⊗ K_{i}K_{i+1}^{−1}),
Δ^{r−1}(F_{i}) =(K_{i}^{−1}K_{i+1} ⊗ ⋯ ⊗ K_{i}^{−1}K_{i+1} ⊗ F_{i}) + (K_{i}^{−1}K_{i+1} ⊗ ⋯ ⊗ K_{i}^{−1}K_{i+1} ⊗ F_{i} ⊗ 1) + ⋯
⋯ + (F_{i} ⊗ 1 ⊗ ⋯ ⊗ 1),
Δ^{r−1}(K_{i}) =K_{i} ⊗ ⋯ ⊗ K_{i},
Δ^{r−1}(K_{i}^{−1}) =K_{i}^{−1} ⊗ ⋯ ⊗ K_{i}^{−1}.

We introduce certain elements of ℚ(v), as follows.

We will define the quantum integer [a], where a is a nonnegative integer, to be

\frac{v^{a} − v^{−a}}{v − v^{−1}} .

We also define quantized factorials by

[a]! := ∏_{k=1}^{a}[k],

and quantized binomial coefficients by

\left[{a \atop b}\right] := \frac{[a]!}{[b]![a − b]!}.

Note that when v is specialised to 1, these become ordinary integers, factorials and binomial coefficients, respectively.

If X is an element of U and c is a nonnegative integer, then the divided power X^{(c)} is defined to be

\frac{X^{c}}{[c]!} .

Sometimes, it is convenient to work with an integral form of U(gl_{n}), which is denoted by U_{A}(gl_{n}), or (when the context is clear) by U. This is an A-algebra which is generated by the elements of U(gl_{n}) given by

E_{i}^{(c)} (1 ≤ i < n, c ∈ ℕ)  (10)
F_{i}^{(c)} (1 ≤ i < n, c ∈ ℕ)  (11)
K_{j} (1 ≤ j ≤ n)  (12)
\left[{K_{j}; 0 \atop t}\right] (1 ≤ j ≤ n, t ∈ ℕ).  (13)

Here,

\left[{K_{i}; c \atop t}\right] := ∏_{s=1}^{t} \frac{K_{i}v^{c−s+1} − K_{i}^{−1}v^{−c+s−1}}{v^{s} − v^{−s}} .

The A-algebra U_{A}^{−} is generated by the elements in (11) subject to relations of form (7) and (9). The A-algebra U_{A}^{0} is generated by the elements in (12) and (13), subject to relations of form

(1) and (2).

The A-algebra U_{A}^{+} is generated by the elements in (10), subject to relations of form (6) and (8). It is known (see [L3, §3.2]) that U ≅ U^{−} ⊗ U^{0} ⊗ U^{+} as A-modules. A good reference for the general theory of quantized enveloping algebras is [L3].

1.1.2 Root systems and root vectors

We now state without proof some properties of root systems. The general theory of these can be found in any good text on Lie algebras.

Associated with the Lie algebra sl_{n}, or (in our case) gl_{n}, is a certain collection of vectors in (n − 1)-dimensional Euclidean space known as a root system of type A_{n−1}. It is well-known that this root system contains an independent subset (the fundamental roots) {α_{1}, …, α_{n−1}} such that any other root is of form

α = α_{i} + α_{i+1} + ⋯ + α_{j} (1 ≤ i ≤ j < n)

or of form

α = −α_{i} − α_{i+1} − ⋯ − α_{j} (1 ≤ i ≤ j < n).

In the first case, the root α is called positive, and in the second case, the root α is called negative. Denote these two sets of roots by Φ^{+} and Φ^{−}, respectively.

We will also write α(i, j + 1) to denote the positive root α_{i} + α_{i+1} + ⋯ + α_{j}.

We define the height, h(α), of α = α(i, j) to be j − i.

Following [L2, §2.2], we define the function g(α(i, j)) = j − 1. (The function g finds the index of the highest fundamental root occurring with nonzero coefficient in its argument.)

The bilinear map ( , ) : Φ^{+} × Φ^{+} → ℤ is defined to satisfy

(α_{i}, α_{j}) := \{ \begin{matrix}2 & if i = j; \\ −1 & if |i − j| = 1; \\ 0 & otherwise. \end{matrix}

Associated to each positive root α in gl_{n}, we define an element E_{α} in U^{+} and an element F_{α} in U^{−}.

Let α = α_{i} + α_{i+1} + ⋯ + α_{j} be a (typical) positive root in type A_{n−1}, where the α_{i} are, as usual, the fundamental roots. If i = j, we define E_{α} := E_{i} and F_{α} := F_{i}. If i ≠ j, we let γ = α − α_{j} and β = α − α_{i} and define, by induction on j − i,

E_{α} := E_{γ}E_{j} − v^{−1}E_{j}E_{γ}

and

F_{α} := F_{β}F_{i} − v^{−1}F_{i}F_{β}.

We also order the elements E_{α} and the elements F_{α} as follows.

The element E_{α(i,j)} precedes (or appears to the left of) the element E_{α(k,l)} if i > k or (i = k and j > l). We denote by β_{N}, …, β_{2}, β_{1} the sequence of positive roots corresponding to this sequence of root vectors.

The element F_{α(i,j)} precedes (or appears to the left of) the element F_{α(k,l)} if j < l or (j = l and i < k). We denote by γ_{N}, …, γ_{2}, γ_{1} the sequence of positive roots corresponding to this sequence of root vectors.

Example Let n = 4. In this case, the positive roots are

α_{1}, α_{1} + α_{2}, α_{1} + α_{2} + α_{3}, α_{2}, α_{2} + α_{3}, α_{3}.

The ordering on the elements E_{α} corresponds to the ordering

α(3, 4) < α(2, 4) < α(2, 3) < α(1, 4) < α(1, 3) < α(1, 2)

on the positive roots, and the ordering on the elements F_{α} corresponds to the ordering

α(1, 2) < α(1, 3) < α(2, 3) < α(1, 4) < α(2, 4) < α(3, 4)

on the positive roots.

1.1.3 Basis theorems for Quantized Enveloping Algebras

In this section, we work with the integral form U = U_{A}. The aim is to define a Poincaré-Birkhoff-Witt type basis for U by using the elements E_{α} and F_{α} which were introduced in §1.1.2. Lemma 1.1

Define ψ^{+} : U^{+} → U^{+} by ψ^{+}(E_{i}) := E_{n−i}. Then ψ^{+} extends naturally to an A-algebra isomorphism ψ^{+} : U^{+} → U^{+}.

Define ψ^{−} : U^{−} → U^{−} by ψ^{−}(F_{i}) := F_{n−i}. Then ψ^{−} extends naturally to an A-algebra isomorphism ψ^{−} : U^{−} → U^{−}. Proof

The map ψ^{+} is self-inverse, and preserves the relations (6) and (8). The map ψ^{−} is self-inverse, and preserves the relations (7) and (9).

Lemma 1.2 There is an A-algebra isomorphism ω^{−} : U^{−} → U^{+} given by ω^{−}(F_{i}) = E_{i}.

There is an A-algebra isomorphism ω^{+} : U^{+} → U^{−} given by ω^{+}(E_{i}) = F_{i}.

Proof Since ω^{−} and ω^{+} are mutual inverses, and they are clearly surjective, it suffices to check that each one preserves the relations. This is immediate from the nature of the relations (6), (7),

(8) and (9).

Lemma 1.3

(i) ψ^{+}(ω^{−}(F_{α(i,j)})) = E_{α(n+1−j,n+1−i)}. (ii) F_{α} precedes F_{β} in the ordering on the elements F_{γ} if and only if ψ^{+}(ω^{−}(F_{α})) precedes

ψ^{+}(ω^{−}(F_{β})) in the ordering on the E_{γ}. Proof

We first prove (i), using induction on h = h(α). The case h = 1 follows from the definition of ω^{−}.

For the general case, F_{α(i,j)} = F_{α(i+1,j)}F_{α(i,i+1)} − v^{−1}F_{α(i,i+1)}F_{α(i+1,j)}, by definition. By induction, we have

ψ^{+}(ω^{−}(F_{α(i,j)})) = E_{α(n−j+1,n−i)}E_{α(n−i,n−i+1)} − v^{−1}E_{α(n−i,n−i+1)}E_{α(n−i,n−j+1)},

because ψ^{+}ω^{−} is an algebra isomorphism. The result now follows from the defintion of

E_{α(n−j+1,n−i+1)}.

The proof of (ii) is immediate from the claim of (i) and the definitions of the two orders.

Definition

Define V^{−} to be the A-algebra given by generators {F̂_{α}^{(c)} : α ∈ Φ^{+}, c ∈ ℤ_{≥0}} (where F_{α}^{(0)} = 1) and relations

F̂_{α}^{(c)}F̂_{α}^{(b)} = \left[{c + b \atop c}\right] F̂_{α}^{(c+b)},  (14)
F̂_{α_{i}}^{(c)}F̂_{α}^{(b)} = F̂_{α}^{(b)}F̂_{α_{i}}^{(c)} if (α, α_{i}) = 0 and i < g(α), (15)
F̂_{α}^{(b)}F̂_{α′}^{(c)} = ∑_{j≥0;j≤c;j≤b} v^{−j−(c−j)(b−j)}F̂_{α′}^{(c−j)}F̂_{α+α′}^{(j)} F̂_{α}^{(b−j)},  (16)
v^{cb}F̂_{α′}^{(c)}F̂_{α+α′}^{(b)} = F̂_{α+α′}^{(b)} F̂_{α′}^{(c)},  (17)
v^{cb}F̂_{α+α′}^{(b)} F̂_{α}^{(c)} = F̂_{α}^{(c)}F̂_{α+α′}^{(b)}.  (18)

The relations (16), (17) and (18) are each subject to the restrictions that (α, α′) = −1 and either (α′ = α_{i} and i < g(α)) or (h(α′) = h(α) + 1 and g(α′) = g(α)).

Proposition 1.4 (Lusztig)

There is an A-algebra isomorphism ϕ : V^{−} → U^{−} satisfying ϕ(F̂_{α_{i}}) = F_{i}, and the set

{ ∏_{α∈Φ^{+}} F̂_{α}^{(c_{α})} : c_{α} ∈ ℤ_{≥0}}

is an A-basis for V^{−}, where the order taken for the product is the same as the order on our elements F_{α}. Proof

The required isomorphism is exhibited in [L2, Theorem 4.5]. This theorem also shows that with a certain fixed order, the products as shown above form an A-basis for V^{−}. Fortunately, this fixed order (which is the reverse of the order shown in [L2, 2.9 (a)]) is exactly the same as the order we imposed on the elements F_{α}!

From [L2, Corollary 4.3] and [L2, Proposition 1.8 (d)], we see that ϕ(F̂_{α_{i}}) = F_{i}. This completes the proof.

Proposition 1.5

(i) The set

B^{−} := { ∏_{α∈Φ^{+}} F_{α}^{(c_{α})} : c_{α} ∈ ℤ_{≥0}} ,

where the product is taken in the order corresponding to that on the elements F_{α}, is an A-basis for U^{−}.

(ii) The set

B^{+} := { ∏_{α∈Φ^{+}} E_{α}^{(c_{α})} : c_{α} ∈ ℤ_{≥0}} ,

where the product is taken in the order corresponding to that on the elements E_{α}, is an A-basis for U^{+}.

Proof The result (ii) will follow from Lemma 1.3 and (i), so it is enough to prove (i). To prove (i), notice that the relation (16) shows that

v^{−1}F̂_{α(i,j)} = F̂_{α(i+1,j)}F̂_{α(i,i+1)} − v^{−1}F̂_{α(i,i+1)}F̂_{α(i+1,j)} Since we know that F_{α(i,i+1)} = F̂_{α(i,i+1)}, we now see by an induction on h(α) that F_{α} =

v^{−h(α)+1}F̂_{α}. Since the claim of (i) is true if we replace F by F̂ whenever it appears in the statement, and the element F_{α} differs from F̂_{α} by a unit in A, we see that (i) holds.

§1.2 Schur Algebras and q-Schur Algebras

1.2.1 The Classical Schur Algebra

Denote by Θ_{r} the set of n × n matrices with nonnegative integer coefficients whose entries sum to r.

The classical Schur algebra, S(n, r) has a basis consisting of certain elements ξ_{i,j} which are defined (as in [G1, §2.3]) as follows.

Let I(n, r) be the set of all ordered r-tuples of elements from the set n := {1, …, n}. The symmetric group S_{r} acts on the set I = I(n, r) on the right by place permutation via

(i_{1}, …, i_{r}).π := (i_{π(1)}, …, i_{π^{(}r)}).

It also acts on the set I × I as (i, j)π = (iπ, jπ). We write i ∼ j if i and j are in the same S_{r}-orbit of I, and (i, j) ∼ (i′, j′) if (i, j) and (i′, j′) are in the same S_{r}-orbit of I × I. The subscripts i and j of the element ξ_{i,j} lie in I(n, r), and we identify ξ_{i,j} and ξ_{i′,j′} if and only if (i, j) ∼ (i′, j′). The set of all ξ_{i,j}, as (i, j) ranges over a transversal Ω of all S_{r}-orbits of I × I can be shown (see e.g. [G1]) to be a basis for S(n, r). We will usually write ξ_{l} as shorthand for ξ_{l,l}.

Schur’s product rule for the basis {ξ_{i,j}} is given by

ξ_{i,j}ξ_{k,l} = ∑_{p,q∈Ω} Z(i, j, k, l, p, q)ξ_{p,q},

where Z(i, j, k, l, p, q) := |{s ∈ I : (i, j) ∼ (p, s) and (k, l) ∼ (s, q)}|.

In this thesis, we will often work with another form of the above basis for S(n, r), this time indexed by Θ_{r}. A typical basis element will be denoted by e_{A}, where A is the matrix in question.

Let i = (i_{1}, …, i_{r}) and j = (j_{1}, …, j_{r}) be elements of I. The identification between the two forms of the basis is given by

ξ_{i,j} = e_{A},

where the (x, y)-entry of the matrix A is given by the number of pairs (i_{s}, j_{s}) such that i_{s} = x and j_{s} = y. It is easily seen that this is well-defined and that the matrix A lies in Θ_{r}.

1.2.2 The q-Schur Algebra Let V be a vector space of dimension r over a field F, and let F be the set of all n-step flags

V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n} = V.

The group G = GL(V) acts naturally on F, hence on X = F × F. Choose (f, f′) ∈ X. Then

f = (V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n}), and f′ = (V′_{1} ⊂ V′_{2} ⊂ ⋯ ⊂ V′_{n}).

Set V_{0} = V′_{0} = {0} and define

a_{ij} = dim(V_{i−1} + (V_{i} ∩ V′_{j})) − dim(V_{i−1} + (V_{i} ∩ V′_{j−1})).

The map from (f, f′) to (a_{ij}) induces a bijection between the set of G orbits on X and the set Θ_{r} (see [D4]). Define O_{A} to be the G-orbit corresponding to A ∈ Θ_{r}.

Now suppose F as above is a finite field with q elements. It is shown in [BLM] that for A, A′, A′′ ∈ Θ_{r}, there exists a function g_{A,A′,A′′,q} given by

g_{A,A′,A′′,q} := |{f ∈ F : (f_{1}, f) ∈ O_{A}, (f, f_{2}) ∈ O_{A′}}| = c_{0} + c_{1}q + ⋯ + c_{m}q^{m},

where the c_{i} are integers that do not depend on the prime power q, and (f_{1}, f_{2}) ∈ O_{A′′}. The ℤ[v^{2}]-polynomial g_{A,A′,A′′} is defined by

g_{A,A′,A′′} := c_{0} + c_{1}v^{2} + ⋯ + c_{m}v^{2m}.

We now define (following [D4] or [BLM, Proposition 1.2]) the q-Schur algebra, S_{q}(n, r), to be the ℚ(v)-vector space with basis {e_{A} : A ∈ Θ_{r}} with associative multiplication given by

e_{A}e_{A′} = ∑_{A′′∈Θ_{r}} g_{A,A′,A′′}e_{A′′}.

Du [D4] remarks that this algebra is canonically isomorphic to the q-Schur algebra defined by Dipper and James, by exhibiting the following correspondence between basis elements e_{A} and basis elements ϕ_{λμ}^{d} as defined by Dipper and James in [DJ3]. Here, the elements λ and μ lie in Λ(n, r), which is the set of compositions of r into n parts, and d ∈ D_{λμ}, which is the set of distinguished W_{λ}–W_{μ} double coset representatives for the Young subgroups W_{λ} and W_{μ} of S_{r}, the symmetric group on r letters. Suppose λ = (λ_{1}, …, λ_{n}) and μ = (μ_{1}, …, μ_{n}). For each α ∈ n, we define I_{α} to be the subset of n given by

I_{α} := {λ_{1} + ⋯ + λ_{α−1} + 1, λ_{1} + ⋯ + λ_{α−1} + 2, …, λ_{1} + ⋯ + λ_{α}}.

Similarly, we define J_{α} to be the analogous subset of n corresponding to μ and α. Given a Dipper-James basis element, ϕ_{λμ}^{d}, we define a corresponding matrix A via

A_{αβ} := |d(J_{β}) ∩ I_{α}|.

This procedure sets up the required isomorphism by sending ϕ_{λμ}^{d} to e_{A}. It should be noted that λ corresponds to the sums of the rows of A, and μ to the sums of the columns of A.

We will also be using certain elements [A] in S_{v}(n, r). These are closely related to the basis elements e_{A} via

[A] := v^{− dim O_{A}+dim pr_{1}(O_{A})}e_{A}.

Here, the map pr_{1} is the first projection from X to F. Beilinson et al. [BLM, 2.3] prove that

dim O_{A} − dim pr_{1}(O_{A}) = ∑_{i,j,k,l} A_{ij}A_{kl},

where the indices are required to satisfy i ≥ k and j < l.

For i, j ∈ I(n, r), we saw in §1.2.1 that ξ_{i,j} = e_{A}, where ξ_{i,j} ∈ S(n, r). We now define the corresponding element ξ_{i,j} ∈ S_{q}(n, r) as the element e_{A} ∈ S_{q}(n, r), where A is as above. Thus the set {ξ_{i,j}}, as (i, j) ranges over a transversal Ω of orbits of I × I, is a basis for S_{q}(n, r). (We are using the notation ξ_{i,j} for an element of S_{q}(n, r) and for an element of S(n, r), but it will be clear which is meant from context.)

The following well-known facts about the multiplication in S_{q}(n, r) are important. (Proofs can be found in [DJ3, §2].)

(i) ξ_{i,j}ξ_{k,l} = 0 unless j ∼ k, in which case it is nonzero. (ii) ξ_{i}ξ_{i,j} = ξ_{i,j}. (iii) ξ_{i,j}ξ_{j} = ξ_{i,j}.

1.2.3 The connection between q-Schur Algebras and Quantized Enveloping Algebras

One of the main aims of this thesis is to investigate the relationship between q-Schur algebras and quantized enveloping algebras.

Beilinson et al. [BLM, §5.7] define a surjective algebra homomorphism θ : U → S_{v}(n, r). This makes S_{v}(n, r) into a U-module, and shows that it is a quotient of U. This is given, following [D4], as follows:

\begin{aligned}θ(E_{i}) & = ∑_{D∈𝔻,E_{i,i+1}+D∈Θ_{r}}[E_{i,i+1} + D], \\ θ(F_{i}) & = ∑_{D∈𝔻,E_{i+1,i}+D∈Θ_{r}}[E_{i+1,i} + D], \\ θ(K_{i}) & = ∑_{D∈𝔻_{r}} v^{d_{i}}[D], \\ θ(K_{i}^{−1}) & = ∑_{D∈𝔻_{r}} v^{−d_{i}}[D]. \end{aligned}

Here, 𝔻 is the set of diagonal matrices, and 𝔻_{r} means 𝔻 ∩ Θ_{r}. The matrix E_{a,b} has 1 in the (a, b) position and zeros elsewhere, and the matrix D is of the form diag(d_{1}, …, d_{n}).

§1.3 Coxeter Systems and their Properties

In the course of this thesis we will be working with symmetric groups and their associated Hecke algebras. These objects arise from Coxeter systems of type A. We therefore introduce in this section the properties of general Coxeter systems.

1.3.1 Coxeter Groups and Hecke Algebras

Following [H, p. 105], we define a Coxeter system to be a pair (W, S) consisting of a group W and a set of generators S ⊂ W, subject only to relations of the form

(ss′)^{m(s,s′)} = 1,

where m(s, s) = 1 and m(s, s′) = m(s′, s) ≥ 2 for s ≠ s′ in S. When the presentation is understood, we shall refer to W as a Coxeter group.

We shall be particularly concerned with the Coxeter group of type A_{r−1}. This can be thought of as the symmetric group on r letters, S_{r}. As a Coxeter group, it is given by generators s_{1}, …, s_{r−1}, where m(s_{i}, s_{j}) = 2 if |i − j| > 1 and m(s_{i}, s_{j}) = 3 if |i − j| = 1. The element s_{i} is identified with the simple transposition (i, i + 1).

Define the length of w ∈ W, denoted by ℓ(w), to be the smallest r for which an expression w = s_{i_{1}} ⋯ s_{i_{r}} exists. Such a minimal expression for w is called a reduced expression.

From any Coxeter system with a finite Coxeter group, we construct the so called Hecke algebra, denoted by H(W). This has basis elements T_{w} parametrised by w ∈ W, and the following multiplication rules:

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) > ℓ(w); \\ qT_{sw} + (q − 1)T_{w} & if ℓ(sw) < ℓ(w). \end{matrix}

(The symbol q appearing in this definition is an indeterminate, as explained earlier.) The existence of this algebra structure is proved in [H, §7]. It follows from the relations that if w = s_{i_{1}} ⋯ s_{i_{r}} is reduced, then T_{w} = T_{s_{1}} ⋯ T_{s_{r}}. It also

follows that if ℓ(uw) = ℓ(u) + ℓ(w) then T_{uw} = T_{u}T_{w}.

When q is replaced by 1, it is known that the algebra H = H(W) is canonically isomorphic to the group algebra of W.

We will require the following property of Hecke algebras: Lemma 1.6 Let d ∈ W. Then

T_{d}. ( ∑_{w∈W} T_{w}) = q^{ℓ(d)}. ( ∑_{w∈W} T_{w}) .

Proof This is well-known (see [DJ1, §3.2]) and follows by induction on ℓ(d).

1.3.2 Parabolic Subgroups and Young Subgroups

Consider a Coxeter system (W, S). For any subset I ⊂ S, we can define a subgroup W_{I} of W, generated by the set I. This is called a parabolic subgroup of W. The following properties of parabolic subgroups are proved in [H, Theorem 5.5].

(i) For each subset I of S, the pair (W_{I}, I) with the given values m(s, s′) is a Coxeter system. Let ℓ_{I} denote the length function for this Coxeter system.

(ii) Let I ⊂ S. If w = s_{i_{1}} ⋯ s_{i_{r}} (s_{i} ∈ S) is a reduced expression, and w ∈ W_{I}, then all s_{i} ∈ I. In

particular, the function ℓ agrees with ℓ_{I} on W_{I}, and W_{I} ∩ S = I. (iii) The assignment I ↦ W_{I} defines a lattice isomorphism between the collection of subsets of S

and the collection of subgroups W_{I} of W. (iv) S is a minimal generating set for W.

Let λ ∈ Λ(n, r). We can associate to λ a certain parabolic subgroup known as the Young subgroup. This is defined to be the subgroup S_{λ} of S_{r} which is generated by all simple transpositions (p, p + 1) such that p and p + 1 are in the same segment of the composition of r given by λ.

1.3.3 The Length Function for Symmetric Groups

The length function for a general Coxeter system was defined in §1.3.1. Dipper and James [DJ1, §1] recall an alternate definition of the length function, applicable to symmetric groups, which is formulated rather differently. Suppose w ∈ S_{r}. They define

ℓ(w) := {(i, j) : i < j and w(i) > w(j)}.

They also state the following useful identities:

ℓ(s_{i}w) = \{ \begin{matrix}ℓ(w) + 1 & if w(i) < w(i + 1); \\ ℓ(w) − 1 & if w(i) > w(i + 1); \end{matrix}
ℓ(ws_{i}) = \{ \begin{matrix}ℓ(w) + 1 & if w^{−1}(i) < w^{−1}(i + 1); \\ ℓ(w) − 1 & if w^{−1}(i) > w^{−1}(i + 1). \end{matrix}

1.3.4 Distinguished Coset Representatives

Let W_{I} be a parabolic subgroup of the Coxeter group W, corresponding to the subset I ⊂ S. We now define

D_{I} := {w ∈ W : ℓ(sw) > ℓ(w) for all s ∈ I}.

By using a reversed version of [H, Proposition 1.10], we find that for any w ∈ W, there exists a unique a ∈ W_{I} and b ∈ D_{I} such that w = ab and ℓ(w) = ℓ(a) + ℓ(b). The element b is the unique right coset representative of W_{I} of shortest length in W, and is called the distinguished right coset representative of w (relative to W_{I}).

1.3.5 Poincaré Polynomials

Associated with each finite Coxeter system (W, S) is a polynomial W (t) known as the Poincaré polynomial. This is defined as follows.

Set a_{n} := |{w ∈ W : ℓ(w) = n}|. Then

W (t) := ∑_{n≥0} a_{n}t^{n} = ∑_{w∈W} t^{ℓ(w)}.

It is convenient to introduce at this point two vector spaces (known as tensor space and tensor matrix space) which play key rôles in this thesis. They will both turn out to be right modules for the Hecke algebra of S_{r}, although we will not show this for tensor matrix space until later.

1.3.6 Tensor Space

Let V be the ℚ(v)-vector space with basis (e_{1}, …, e_{n}). Then T^{r}(V) = V^{⊗r} can be made into a U–H bimodule as follows. Firstly we can make V into a left U-module via

\begin{aligned}E_{i}e_{i+1} & = e_{i}, \\ E_{i}e_{j} & = 0 (j ≠ i + 1), \\ F_{i}e_{i} & = e_{i+1}, \\ F_{i}e_{j} & = 0 (j ≠ i), \\ K_{i}e_{i} & = ve_{i}, \\ K_{i}e_{j} & = e_{j} (j ≠ i). \end{aligned}

We now make T^{r}(V) into a left U-module via the homomorphism Δ^{r−1}. Following [D4], we now make T^{r}(V) into a right H-module via

e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}T_{p,p+1} := \{ \begin{matrix}ve_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} < i_{p+1}; \\ v^{2}e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} = i_{p+1}; \\ ve_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}} + (v^{2} − 1)e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}} & if i_{p} > i_{p+1}. \end{matrix}

Jimbo [J] proved that there exists a quotient ρ_{r}(U) of U such that ρ_{r}(U) = End_{H}(T^{r}(V)) and H = End_{U}(T^{r}(V)) when n ≥ r. It is shown in [DJ3, Theorem 6.6] that the q-Schur algebra S_{q}(n, r) is the centralising algebra of H with respect to “q-tensor space”. It is shown in [D4] that Dipper and James’ so-called q-tensor space is essentially the same as the classical tensor space defined above, but with certain powers of v introduced. Since the rings of H-endomorphisms of the two spaces are isomorphic, we may work with either, and in this thesis, we will mainly be using classical tensor space. Thus, the v-Schur algebra can be identified with the quotient of U which acts faithfully on T^{r}(V).

Sometimes it will be more convenient to write u_{i} for e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}, where i is the element of I(n, r) given by (i_{1}, …, i_{r}).

We define the function m : I(n, r) → ℤ by

m(i) := |{⟨a, b⟩ : 1 ≤ a < b ≤ r : i_{a} > i_{b}}|.

We will write [u_{i}] for v^{−m(i)}u_{i}.

The action of H can now be re-expressed as follows. Let s = (p, p + 1). Then

[u_{i}].T_{s} = \{ \begin{matrix}q[u_{i.s}] & if i_{p} ≤ i_{p+1}; \\ [u_{i.s}] + (q − 1)[u_{i}] & if i_{p} > i_{p+1}. \end{matrix}

The action of T_{s}^{−1} is also easy to describe:

[u_{i}].T_{s}^{−1} = \{ \begin{matrix}q^{−1}[u_{i.s}] & if i_{p} ≥ i_{p+1}; \\ [u_{i.s}] + (q^{−1} − 1)[u_{i}] & if i_{p} < i_{p+1}. \end{matrix}

1.3.7 Tensor Matrix Space

Note that M_{n} = M_{n}(R), the n × n matrix algebra over R (where R = ℚ for the classical case and R = ℚ(v) for the quantized case), has a natural U-module structure as follows. Denote by e_{ij} or e_{i,j} the matrix unit with 1 in the (i, j)-position and zeros elsewhere. A quick check shows that there is an algebra homomorphism θ_{1} from U to M_{n} satisfying:

\begin{aligned}θ_{1}(E_{i}) & = e_{i,i+1}, \\ θ_{1}(F_{i}) & = e_{i+1,i}, \\ θ_{1}(K_{i}) & = ∑_{j=1}^{n} v^{δ(i,j)}e_{j,j}, \\ θ_{1}(K_{i}^{−1}) & = ∑_{j=1}^{n} v^{−δ(i,j)}e_{j,j}. \end{aligned}

We will regard M_{n} as a left U-module via the action

u.m := θ_{1}(u) × m.

We define tensor matrix space to be T^{r}(M_{n}), the r-th tensor power of M_{n}. This vector space has an associative multiplication defined on it given by

e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}} × e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}} := \{ \begin{matrix}e_{a_{1}d_{1}} ⊗ ⋯ ⊗ e_{a_{r}d_{r}} & if b_{i} = c_{i} for all i; \\ 0 & otherwise. \end{matrix}

We regard T^{r}(M_{n}) as a U-module via the comultiplication Δ.

Sometimes it will be more convenient to write u_{i,j} for

e_{i_{1},j_{1}} ⊗ ⋯ ⊗ e_{i_{r},j_{r}},

where i, j ∈ I(n, r) are given by (i_{1}, …, i_{r}) and (j_{1}, …, j_{r}) respectively.

The symmetric group S_{r} acts on T^{r}(M_{n}) on the right by place permutation of the tensors, which corresponds, in the u_{i,j} notation, to the action

u_{i,j}.π = u_{i.π,j.π},

where the action of S_{r} on I(n, r) is as in §1.2.1.

We say that a typical basis element u_{i,j} of T^{r}(M_{n}) is is of parabolic form if the stabilizer of j under the action of S_{r} on I(n, r) is a parabolic subgroup. We will also say in this case that the element j is of parabolic form. An equivalent way of looking at this is that if 1 ≤ a < b < c ≤ r and j_{a} = j_{c}, then j_{a} = j_{b} = j_{c}.

§1.4 Combinatorics

In this section we introduce some combinatoric definitions which will be important for later purposes.

1.4.1 Two Orders

Let A and B be ordered sets. Then the Hebrew lexicographic order on A × B is given by ⟨a, b⟩ < ⟨c, d⟩ if and only if b < d or (b = d and a < c). The elements of a sequence (i_{1}, …, i_{m}) in A × B are said to appear in Hebrew lexicographic order if i_{1} ≤ ⋯ ≤ i_{r} with respect to this order.

A basis element u_{i,j} of T^{r}(M_{n}) is said to be H-relaxed if the elements of the sequence

(⟨i_{1}, j_{1}⟩, …, ⟨i_{r}, j_{r}⟩)

appear in Hebrew-lexicographic order.

Let A and B be ordered sets. Then the relaxed order on A × B is given by ⟨a, b⟩ < ⟨c, d⟩ if and only if b < d or (b = d and a > c). The elements of a sequence (i_{1}, …, i_{m}) in A × B are said to appear in relaxed order if i_{1} ≤ ⋯ ≤ i_{r} with respect to this order.

A basis element u_{i,j} of T^{r}(M_{n}) is said to be relaxed if the elements of the sequence

(⟨i_{1}, j_{1}⟩, …, ⟨i_{r}, j_{r}⟩)

appear in the relaxed order.

Examples Let j = (1, 1, 1, 1, 3, 3, 3, 7, 7, 7).

If i = (1, 4, 6, 7, 2, 4, 5, 3, 3, 4) then u_{i,j} is H-relaxed.
If i′ = (7, 6, 4, 1, 5, 4, 2, 4, 3, 3) then u_{i′,j} is relaxed.

1.4.2 Tableaux and Codeterminants We now make some combinatorial definitions, most of which appear in [G2]. We define a codeterminant to be a nonzero product of two basis elements ξ_{a,b}ξ_{c,d} of S(n, r).

(The codeterminant is determined by the given factorisation, and not by the element of S(n, r) which is equal to it.)

We identify Λ(n, r), the set of compositions of r into at most n pieces, with the set

Λ = {λ = (λ_{1}, …, λ_{n}) : λ_{ν} ∈ ℕ_{0} for all ν ∈ n and ∑_{ν=1}^{n} λ_{ν} = r}.

We also define

Λ^{+} = Λ^{+}(n, r) := {λ ∈ Λ(n, r) : λ_{1} ≥ ⋯ ≥ λ_{n}}.

(Here, ℕ_{0} is the set of natural numbers, including 0.)

An element of Λ is called a weight, and is dominant if λ ∈ Λ^{+}. There is an obvious correspondence between elements of Λ^{+} and partitions of r into not more than n parts.

The lexicographic order on Λ is defined as follows. μ ≻ λ if for some c, μ_{a} = λ_{a} for all a < c, and μ_{c} > λ_{c}.

The weight wt(i) of an element i ∈ I(n, r) is the element α ∈ Λ given by α_{ν} = |{ρ ∈ r : i_{ρ} = ν}| for all ν ∈ n. If i, j ∈ I, it is clear that i ∼ j if and only if wt(i) = wt(j).

For each λ ∈ Λ^{+} we define a basic λ-tableau T^{λ} by writing the integers 1, …, r into a Young diagram in some arbitrary (but henceforth fixed) order. (In practice, the order we pick will always be row by row, starting with the top row, and filling each row from left to right.) To each i ∈ I we now associate the λ-tableau T_{i}^{λ} = iT^{λ}. For example, let n = 4, r = 7, λ = (4, 2, 1, 0). Using our choice of basic λ-tableau, then

T^{λ} = 1 5 7 2 6 3 4 ,

and

T_{i}^{λ} = i_{1} ; i_{5} ; i_{7} i_{2} ; i_{6} i_{3} i_{4} .

If λ ∈ Λ^{+} and i ∈ I, the λ-tableau T_{i}^{λ} is said to be standard if the elements in each row increase weakly from left to right, and the elements in each column increase strictly from top to bottom. We define

I_{λ} := I_{λ}(n, r) = {i ∈ I : T_{i}^{λ} is standard}.

We say a λ-tableau T_{i}^{λ} is row-semistandard if the elements in each row increase weakly from left to right, and define the set

I′_{λ} := I′_{λ}(n, r) = {i ∈ I : T_{i}^{λ} is row − semistandard}.

It is clear that in a standard tableau, all the entries equal to s must appear in the first s rows, by an easy induction on s. There is exactly one element of I, denoted by ℓ = ℓ(λ), for which T_{ℓ}^{λ} is standard and wt(ℓ) = λ. (The entries in the s-th row of the tableau T_{ℓ}^{λ} are all equal to s.)

The elements of I can be lexicographically ordered as follows: i ⪰ j if and only if i = j or, if ρ is minimal subject to the condition that i_{ρ} ≠ j_{ρ}, then i_{ρ} < j_{ρ}.

If λ ∈ Λ^{+}, we define the element λ′ ∈ Λ^{+} by the rule that λ′_{h} is the number of elements in column number h of the tableau of shape λ.

Example Suppose n = 4, r = 9 and λ = (4, 3, 2, 0). Then a tableau of shape λ is of the form

.

In this case, the dual tableau λ′ is given by (3, 3, 2, 1) and corresponds to the array given by

.

It is clear for any p ∈ I that p ∼ ℓ(wt(p)). Thus, any basis element ξ_{a,b} of S(n, r) can be written as ξ_{i,ℓ(μ)} where μ = wt(b), or as ξ_{ℓ(ν),j} where ν = wt(a). The product ξ_{a,b}ξ_{c,d} will be zero unless b ∼ c, and hence wt(b) = wt(c), so it is easy to see that a general codeterminant can be written as

Y_{i,j}^{λ} := ξ_{i,ℓ(λ)}ξ_{ℓ(λ),j},

where i, j ∈ I and λ ∈ Λ. We say that Y_{i,j}^{λ} is a codeterminant of shape λ. If, in addition, λ ∈ Λ^{+} and i and j lie in I_{λ}, we call the codeterminant Y_{i,j}^{λ} a standard codeterminant. It should be noted that if b ∼ c then ξ_{a,b}ξ_{c,d} is nonzero. This is a corollary of Schur’s product rule.

We define a q-codeterminant to be a product e_{A}e_{A′} ∈ S_{q}(n, r) such that the corresponding product e_{A}e_{A′} of S(n, r) is a codeterminant. (The q-codeterminant is determined by the matrices A and A′, and not solely by the element of S_{q}(n, r) which is equal to it.) We also define a vcodeterminant to be an element [A][A′] of S_{v}(n, r), where A and A′ are as above. Standard qcodeterminants and standard v-codeterminants are defined in the obvious way. It is not hard to see that when q and v are replaced by 1, the q-codeterminants and v-codeterminants specialise to the classical codeterminants introduced above.

It is convenient to introduce the following elements of ℤ[q], as in [DJ1].

a) For an nonnegative integer n, [n]_{q} := 1 + q + ⋯ + q^{n−1}.

b) Define [n]_{q}! := [1]_{q}[2]_{q} ⋯ [n]_{q}. Let a, b ∈ I(n, r). Then the element P_{a,b} ∈ ℤ[q] is defined to be

∏_{i,j=1}^{n}[c_{ij}]_{q}!,

where c_{ij} is the multiplicity of ⟨i, j⟩ in the family {⟨a_{1}, b_{1}⟩, …, ⟨a_{r}, b_{r}⟩}. If t = u_{a,b} then P_{t} := P_{a,b}.

In [DJ3, §3], the element w_{λ} ∈ S_{r} is introduced. It is defined as the distinguished S_{λ}–S_{λ′} double coset representative with the property that

w_{λ}^{−1}S_{λ}w_{λ} ∩ S_{λ′} = 1.

The important point to notice is that when the element of I(n, r) corresponding to T^{λ}.w_{λ} is written into a tableau of shape λ′ in the usual way, then it is the transpose of T^{λ}. For example, if T^{λ} is given by

1 5 8 2 6 9 3 7 4

then, as a tableau of shape λ′, the element (1, 2, 3, 4, 5, 6, 7, 8, 9).w_{λ} is given by

1 2 3 4 5 6 7 8 9 .

1.4.3 Dominance Order

We define the partial order of dominance, ⊴, on Λ^{+}(n, r) by stipulating that λ ◁ μ (μ dominates λ) if

λ_{1} + ⋯ + λ_{s} ≤ μ_{1} + ⋯ + μ_{s}

for all 1 ≤ s ≤ n.

Following [M, §3.1], we call π ⊆ Λ^{+}(n, r) an increasing saturation if for any λ ∈ π and μ ▷ λ we have μ ∈ π.

We define the total order ≻ on Λ(n, r) by μ ⪰ λ if μ = λ or the smallest i such that μ_{i} ≠ λ_{i} satisfies μ_{i} > λ_{i}. It is not hard to see that ⪰ refines ⊵, i.e. that μ ⊵ λ implies μ ⪰ λ.

We label the elements of Λ^{+} λ^{1}, …, λ^{t} where the order is chosen to satisfy the condition λ^{1} ≻ ⋯ ≻ λ^{t}.

2. A New Setting for the q-Schur Algebra

The main aim of this chapter is to describe an embedding (of rings and of U-modules) of the q-Schur algebra into the r-th tensor power T^{r}(M_{n}) of the n×n matrix ring. We achieve this in Theorem 2.2.7. This embedding allows products in the q-Schur algebra to be computed in a straightforward manner, and gives a method for generalising results on S(n, r) to S_{q}(n, r). In particular we shall make use of this embedding in §3 to prove a straightening formula in S_{q}(n, r) which generalises the straightening formula for codeterminants due to Woodcock [W].

The chapter falls into three parts. §2.1 discusses the classical Schur algebra from a new perspective compatible with the structure with the corresponding universal enveloping algebra, U(gl_{n}). §2.2 is devoted to quantum analogues of the results in §2.1. §2.3 uses the techniques of §2.2 to describe a family of subalgebras of the q-Schur algebra.

2.1. The classical Schur algebra as a subspace of tensor matrix space We now show how to embed the classical Schur algebra in T^{r}(M_{n}). With each matrix A ∈ Θ_{r}, we can (uniquely) associate a standard basis element t_{A} of T^{r}(M_{n}),

given by

e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}},

where the ordered pairs ⟨a_{i}, b_{i}⟩ appear in Hebrew lexicographic order, and the multiplicity with which e_{ab} occurs in t_{A} is the entry in the (a, b)-position of A. Recall from §1.3.7 that the symmetric group S_{r} acts on the set of basis elements of T^{r}(M_{n}) on the right. Consider the orbit containing the basis element t_{A}, and let D_{A} be the set of distinguished right coset representatives relative to the stabilizer of t_{A} (which is a parabolic subgroup of W (A_{r−1})).

Example Let A be the matrix

( \begin{matrix}3 & 1 \\ 1 & 2 \end{matrix} ) .

Then t_{A} is given by

e_{11} ⊗ e_{11} ⊗ e_{11} ⊗ e_{21} ⊗ e_{12} ⊗ e_{22} ⊗ e_{22}.

Definition We now define a map ψ : S(n, r) → T^{r}(M_{n}), given by

ψ(e_{A}) := t_{A}. ( ∑_{w∈D_{A}} w) ,

and extended by linearity.

It is important to notice that if A ranges over all elements of Θ_{r}, then ψ(e_{A}) ranges over a basis for the subspace of T^{r}(M_{n}) consisting of all elements t such that t.π = t for all π ∈ S_{r}. For example, if n = 2 and r = 3, and A is the matrix

( \begin{matrix}2 & 1 \\ 0 & 0 \end{matrix} ) ,

then

ψ(e_{A}) = e_{11} ⊗ e_{11} ⊗ e_{12} + e_{11} ⊗ e_{12} ⊗ e_{11} + e_{12} ⊗ e_{11} ⊗ e_{11}.

From this, we see that dim (im ψ) is equal to the number of monomials of degree r in n^{2} commuting variables, i.e. \binom{n^{2}+r−1}{r}.

The following results establish that ψ is a monomorphism of algebras, and of U-modules, where U is the universal enveloping algebra of gl_{n}.

Proposition 2.1.1 The map ψ : S(n, r) → T^{r}(M_{n}) is a monomorphism of U-modules. Proof The preceding discussion of the properties of ψ shows that ψ is injective. It is known that the dimension of S(n, r) is given by

\binom{n^{2} + r − 1}{r},

which is the same as the number of monomials of degree r in n^{2} commuting variables, and hence is equal to dim (im ψ). It suffices to check that the effect of E_{i}, F_{i} and H′_{i} is as expected, because these generate U as an algebra. (Here, H′_{i} denotes the specialisation of

\frac{K_{i} − K_{i}^{−1}}{v − v^{−1}}

in quantum gl_{n}. The element H_{i} ∈ sl_{n} is related to it via H_{i} = H′_{i} − H′_{i+1}.) We tackle first the case of E_{h}. From the definition of θ(E_{h}) and the specialisation of [BLM, Lemma 3.4 (a2)], we see that

E_{h}.e_{A} = ∑_{p:a_{h+1,p}≥1}(a_{h,p} + 1) e_{(A+e_{h,p}−e_{h+1,p})}.

Now consider the case of E_{h}.ψ(e_{A}). Considering the number of different terms in ψ(e_{A}) that give rise, under the action of E_{h}, to a particular term of ψ(e_{B}) (for various matrices B of the correct type), we find that the same formula as above still holds.

The situation for F_{h} is entirely analogous, but uses [BLM, Lemma 3.4 (b2)].

It follows from the definition of θ(K_{i}) that

\frac{K_{h} − K_{h}^{−1}}{v − v^{−1}} .e_{A} = [ ∑_{j=1}^{n} A_{hj}] e_{A},

thus

H′_{h}.e_{A} = ∑_{j=1}^{n} A_{hj} e_{A}.

It is not hard to see that H′_{h} acts on ψ(e_{A}) as exactly the same scalar multiple.

These observations suffice to prove the Proposition.

Before showing that ψ is an isomorphism of algebras, we remark that our “tensor matrix space” is canonically isomorphic to an algebra with basis elements u_{i,j} (as in §1.3.7) which appears in the literature (see [G3, §2]). The multiplication rule for the elements u_{i,j} is as follows:

u_{i,j}u_{k,l} = δ_{j,k}u_{i,l},

where δ is the Kronecker delta. The symmetric group on r letters acts on u_{i,j} via u_{i,j}^{π} = u_{iπ,jπ}. There is a linear map from the algebra generated by the u_{i,j} into T^{r}(M_{n}) determined by

σ(u_{i,j}) = e_{i_{1}j_{1}} ⊗ ⋯ ⊗ e_{i_{r}j_{r}}.

It is immediate that σ is an algebra isomorphism.

Proposition 2.1.2 ψ : S(n, r) → T^{r}(M_{n}) is an algebra monomorphism. Proof We know ψ is injective, so we need to prove it is a ring homomorphism.

Consider a basis element ξ_{i,j} of S(n, r), and let A ∈ Θ_{r} be given by ξ_{i,j} = e_{A} as in §1.2.1. We may assume the pair ⟨i, j⟩ is H-relaxed. We claim that

ψ(ξ_{i,j}) = ∑_{π∈X} u_{i.π,j.π},

where X denotes any transversal of the cosets of the stabiliser of u_{i,j} in S_{r}. This follows by definition of ψ in the case where X is the transversal D_{A} corresponding to t_{A} = u_{i,j}. One can see from the properties of symmetric group actions that the sum

∑_{π∈X} u_{i.π,j.π}

is independent of the transversal X.

It now follows from elementary combinatoric considerations that ψ is a homomorphism, because it respects Schur’s product rule. (A similar observation has been made in [G3, equation (2.3)].)

Remark Note that this expression for the ξ_{i,j} in terms of the u_{i,j} motivates the definition of Schur’s product rule.

The maps Δ and θ have classical counterparts obtained by specialising v and K_{i} (for all i) to be 1. Since no confusion is likely to arise, we use the symbols Δ and θ for both classical and quantized maps.

The preceding results can also be viewed in the context of the map Δ : U → U ⊗ U. We have seen that S(n, 1) is isomorphic as a ring and as a U-module to M_{n}, and the same holds for the quantized version S_{v}(n, 1). Denote by θ_{1} the map from U to M_{n} induced by θ : U → S(n, 1). This leads to the following result. Proposition 2.1.3 The map ψθ agrees with the map

γ_{r} := \underbrace{(θ_{1} ⊗ ⋯ ⊗ θ_{1})}_{r times} Δ^{r−1}

on U. Proof Firstly, observe that both maps are from U to T^{r}(M_{n}). Both maps are algebra homomorphisms, because Δ and θ are known to be, θ_{1} is a special case of θ, and ψ has been shown to be. If we can prove that the images under E_{i}, F_{i}, and H′_{i} are the same under both maps, the Theorem will follow.

We find that

γ_{r}(E_{i}) = ∑_{j=1}^{r} 1 ⊗ ⋯ ⊗ 1 ⊗ e_{ii+1} ⊗ 1 ⊗ ⋯ ⊗ 1,

where the e_{ii+1} occurs in the j-th place in the tensor. However, because

∑_{i=1}^{n} e_{ii} = 1

in M_{n}, we see, by expanding the expression for γ_{r}, that it agrees with that given by ψ(θ(E_{i})). The case of γ_{r}(F_{i}) is entirely analogous. In the case of H′_{i},

γ_{r}(H′_{i}) = ∑_{j=1}^{r} 1 ⊗ ⋯ ⊗ 1 ⊗ e_{ii} ⊗ 1 ⊗ ⋯ ⊗ 1,

where the e_{ii} occurs at the j-th place. Expanding 1 as before, we find that each tensor occurs d_{i} times, where d_{i} is the (i, i)-entry of the corresponding matrix in S(n, r). This is exactly as is required.

Remark We shall see that the above result quantizes, in the sense that something similar works when Δ and θ as above are replaced by their quantum analogues. This yields a quantized version of γ_{r} which, again, is an algebra homomorphism. Because tensor space is made into a U-module via a process equivalent to applying γ_{r}, and because the v-Schur algebra can be described as the faithful quotient of U corresponding to this action, it can be seen that

γ_{r}(U) ≅ S_{v}(n, r),

as an algebra and as a U-module. This isomorphism is not as obvious as in the classical case and it is harder to work out the images of basis elements under γ_{r}, but, we shall see in §2.2 that it can be done, by finding a suitable quantum analogue of ψ. This gives our product rule for elements of the q-Schur algebra.

2.2. The v-Schur algebra as a subspace of tensor matrix space

We have seen in the above sections that the classical Schur algebra S(n, r) embeds in T^{r}(M_{n}) in a natural way compatible with the U-module and algebra structures of both objects. Moreover, the portion of T^{r}(M_{n}) identified with the Schur algebra is invariant under the action of the symmetric group S_{r} acting on the right of T^{r}(M_{n}) by place permutation. Next, we define an action of the Hecke algebra H(S_{r}) (corresponding to the symmetric group on r letters) on T^{r}(M_{n}) on the right. The invariant subspace of T^{r}(M_{n}) corresponding to the “ind” representation of the Hecke algebra (which corresponds to the trivial representation of the symmetric group) will remarkably be seen to have a U-module structure and algebra structure which agree precisely with those of the v-Schur algebra. By specialising the parameters to 1, we recover the classical objects of the previous part of this chapter.

Definition Let

e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}}

be a standard basis element of T^{r}(M_{n}). We define the corresponding quantized basis element to be

[e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}}] := v^{−m}e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}},

where

m := |{⟨i, j⟩ : 1 ≤ i < j ≤ r, a_{i} > a_{j}}| + |{⟨i, j⟩ : 1 ≤ i < j ≤ r, b_{i} > b_{j}}|.

If u_{i,j} corresponds to e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}} in the way described in §2.1, then we write [u_{i,j}] for the quantity v^{∗}u_{i,j} corresponding to [e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}}]. (We use the convention that the symbol ∗ represents some integer.)

Definition The action of H(A_{1}) on T^{2}(M_{n}) is defined as follows. Let [e_{ab} ⊗ e_{cd}] be a typical (quantized) basis element. Then [e_{ab} ⊗ e_{cd}].T_{s} (s being the standard generator of the Coxeter group of type A_{1}) is given by:

[e_{ab} ⊗ e_{cd}].T_{s} := \{ \begin{matrix}q[e_{cd} ⊗ e_{ab}] & if a ≤ c, b ≤ d; \\ [e_{cd} ⊗ e_{ab}] + (q − 1)[e_{ab} ⊗ e_{cd}] & if a > c, b > d; \\ [e_{cd} ⊗ e_{ab}] + (q − 1)[e_{ad} ⊗ e_{cb}] & if a > c, b ≤ d; \\ q[e_{cd} ⊗ e_{ab}] + (q − 1)[e_{ab} ⊗ e_{cd}] − (q − 1)[e_{ad} ⊗ e_{cb}] & if a ≤ c, b > d. \end{matrix}

As usual, we identify q with v^{2}. It remains to be seen that this action is well-defined.

The action of H(A_{r−1}) on the quantized basis elements of T^{r}(M_{n}) is defined entirely analogously, with a typical generator T_{(p,p+1)} acting as in the r = 2 case, but affecting only the p-th and (p+1)-th positions in the tensor.

Lemma 2.2.1 The above action is a well-defined action of H(A_{r−1}) on T^{r}(M_{n}). Proof It is well-known (see for example [H]) that H is given by generators T_{1}, …, T_{r−1} subject to the following relations:

\begin{aligned}T_{s}^{2} & = qT_{e} + (q − 1)T_{s}, \\ T_{s}T_{t} & = T_{t}T_{s} if |s − t| > 1, \\ T_{s}T_{t}T_{s} & = T_{t}T_{s}T_{t} if |s − t| = 1. \end{aligned}

Here, T_{e} is the multiplicative identity in H, and s and t are integers between 1 and r − 1 inclusive.

To check the first of these relations, it is enough to consider the case r = 2. The element T_{s} acts on the space spanned by the ordered basis

[e_{ab} ⊗ e_{cd}], [e_{cd} ⊗ e_{ab}], [e_{cb} ⊗ e_{ad}], [e_{ad} ⊗ e_{cb}],

where we assume that a < c and b < d. With respect to this basis, the right action of T_{s} is given by the matrix

M := ( \begin{matrix}0 & q & 0 & 0 \\ 1 & q − 1 & 0 & 0 \\ 0 & q − 1 & 0 & 1 \\ −(q − 1) & 0 & q & q − 1 \end{matrix} ) .

It is routine to check that M^{2} = qI + (q − 1)M, where I is the identity matrix. There are also three degenerate cases to check.

1. The case a = c, b < d. Consider the basis [e_{ab} ⊗ e_{ad}], [e_{ad} ⊗ e_{ab}]. Using the previous method, we find the action of T_{s} is given by

M := ( \begin{matrix}0 & q \\ 1 & q − 1 \end{matrix} ) .

This matrix satisfies M^{2} = qI + (q − 1)M as required.

2. The case a < c, b = d. This is very similar to case 1., except that the basis to take is

[e_{ab} ⊗ e_{cb}], [e_{cb} ⊗ e_{ab}].

3. The case a = c, b = d. Here, T_{s} acts on [e_{ab} ⊗ e_{ab}] via the ind representation, i.e. T_{s} ↦ q. This satisfies the required relation. Since T^{r}(M_{n}) is spanned by quantized basis elements, each of which falls into exactly one of

the above four classes, we have shown that the first relation, T_{s}^{2} = qT_{e} + (q − 1)T_{s}, always holds.

The second relation (known as a short braid relation), T_{s}T_{t} = T_{t}T_{s} when |s − t| > 1, is an easy case. The action of T_{(p,p+1)} is a sum of certain permutations of coordinates in the matrix units occuring in the p-th and (p + 1)-th places, so it is clear that if |s − t| > 1, the permutations will not interfere with each other, and hence the elements T_{s} and T_{t} will commute with each other.

The last relation (known as a long braid relation) is the trickiest to check. To help in the checking, we recall the action of H on tensor space, as defined by Du [D4] and others.

Let V be, as usual, a vector space over ℚ(v) with basis e_{1}, …, e_{n}. We now define quantized basis elements of T^{r}(V) from the usual basis as follows:

[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}] := v^{−m}e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}},

where

m := |{⟨a, b⟩ : 1 ≤ a < b ≤ r, i_{a} > i_{b}}|.

Following [D4], we see that an action of H on T^{r}(V) can be defined as follows:

[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}].T_{p,p+1} = \{ \begin{matrix}q[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}}] & if i_{p} ≤ i_{p+1}; \\ [e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}}] + (q − 1)[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}] & if i_{p} > i_{p+1}. \end{matrix}

We will exploit the fact that this action resembles the case b ≤ d in our candidate for the action of H on T^{r}(M_{n}).

To deal with the case b > d, we calculate the action of T_{s}^{−1} in each case. This follows easily from the fact that T_{s}^{−1} = (q^{−1} − 1)T_{e} + q^{−1}T_{s} and the original definitions, because T_{e} acts as the identity. The action of T_{s}^{−1} on T^{2}(M_{n}) is as follows:

[e_{ab} ⊗ e_{cd}].T_{s}^{−1} := \{ \begin{matrix}q^{−1}[e_{cd} ⊗ e_{ab}] & if a ≥ c, b ≥ d; \\ [e_{cd} ⊗ e_{ab}] + (q^{−1} − 1)[e_{ab} ⊗ e_{cd}] & if a < c, b < d; \\ [e_{cd} ⊗ e_{ab}] + (q^{−1} − 1)[e_{ad} ⊗ e_{cb}] & if a < c, b ≥ d; \\ q^{−1}[e_{cd} ⊗ e_{ab}] + (q^{−1} − 1)[e_{ab} ⊗ e_{cd}] − (q^{−1} − 1)[e_{ad} ⊗ e_{cb}] & if a ≥ c, b < d. \end{matrix}

It follows that the action of T_{(p,p+1)}^{−1} on T^{r}(V) is given by setting [e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}].T_{p,p+1}^{−1} equal to

\{ \begin{matrix}q^{−1}[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}}] & if i_{p} ≥ i_{p+1}; \\ [e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{p+1}} ⊗ e_{i_{p}} ⊗ ⋯ ⊗ e_{i_{r}}] + (q^{−1} − 1)[e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}] & if i_{p} < i_{p+1}. \end{matrix}

Note the similarity between the action on tensor space and the b ≥ d case in the prospective matrix action here. Also note that all the actions have a “continuity” property: for example in any case where a = c, it does not matter whether this is considered as a ≤ c or a ≥ c.

Comparing the actions of T_{s} on [e_{ab} ⊗ e_{cd}] and on [e_{a} ⊗ e_{c}], we find that they are very similar, provided that b ≤ d. What happens is that the second coordinates in the matrix units are exchanged, and the action on the first coordinates is exactly the same as in the action on tensor space. On the other hand, the action of T_{s}^{−1} on [e_{ab} ⊗ e_{cd}] is similar to the action of T_{s}^{−1} on [e_{a} ⊗ e_{c}], provided that b ≥ d. Again, the second coordinates are exchanged, and the first coordinates are acted upon as in the action on tensor space.

With these observations out of the way, we proceed with the proof. Clearly, it is enough to study the case r = 3. Pick a quantized standard basis element, [e_{ab} ⊗ e_{cd} ⊗ e_{ef}]. It is enough to show that T_{s}T_{t}T_{s} acts in the same way as T_{t}T_{s}T_{t} on this basis element, where s corresponds to the permutation (1, 2) and t corresponds to the permutation (2, 3). The above discussion shows that this is guaranteed, provided b ≤ d ≤ f, because we know that the action on tensor space is well-defined. For the other cases, we need another method. For example, if b > d > f, we know from the discussion that T_{s}^{−1}T_{t}^{−1}T_{s}^{−1} acts in the same way as T_{t}^{−1}T_{s}^{−1}T_{t}^{−1}, and this is equivalent to showing that the long braid relation holds. There are six possible cases. Here is a table showing the correct method of proof in each case. (There is some overlap—in these cases, any of the given methods will work.) For clarity, σ means T_{s} and τ means T_{t}.

Situation  Method of Proof
b ≤ d ≤ f  στσ = τστ
f ≤ d ≤ b  σ^{−1}τ^{−1}σ^{−1} = τ^{−1}σ^{−1}τ^{−1}
d ≤ b ≤ f  σ^{−1}τσ = τστ^{−1}
d ≤ f ≤ b  σ^{−1}τ^{−1}σ = τσ^{−1}τ^{−1}
b ≤ f ≤ d  στσ^{−1} = τ^{−1}στ
f ≤ b ≤ d  στ^{−1}σ^{−1} = τ^{−1}σ^{−1}τ

This suffices to prove that the long braid relation is preserved.

This completes the proof.

It will be important in the proofs which follow that the actions of U(gl_{n}) and H(A_{r−1}) on T^{r}(M_{n}) commute with each other. The action of U on M_{n} is given by:

\begin{aligned}E_{i}e_{jk} & := δ_{i,j−1}e_{j−1k}, \\ F_{i}e_{jk} & := δ_{i,j}e_{j+1k}, \\ K_{i}e_{jk} & := v^{δ_{i,j}}e_{jk}, \\ K_{i}e_{jk} & := v^{−δ_{i,j}}e_{jk}, \end{aligned}

where δ is the Kronecker delta. The action extends to T^{r}(M_{n}) via Δ^{r−1}.

Lemma 2.2.2 The actions of H(A_{r−1}) and U on T^{r}(M_{n}) commute with each other. Proof We will rely heavily on the well-known fact that the actions of U and H on tensor space commute with each other. Note that our action of U on T^{r}(M_{n}) is essentially the action of U on tensor space, but ignoring the second coordinate.

Clearly the action of U commutes with the action of T_{e}. It is now enough, for each basis element of T^{r}(M_{n}) and for each Hecke algebra generator T_{s} (corresponding to a simple reflection in the Coxeter group), to establish either that the action of each E_{i}, F_{i}, K_{i} and K_{i}^{−1} commutes with the action of T_{s} or the action of T_{s}^{−1} (because T_{s}^{−1} = (q^{−1} − 1)T_{e} + q^{−1}T_{s}). A moment’s thought shows that it is enough to consider the case r = 2, because only simple transpositions are involved.

Denote the typical basis element by e_{ab} ⊗e_{cd} (note that we choose to work with the unquantized basis of T^{r}(M_{n})). We first dispose of the easy cases of the K_{i} and K_{i}^{−1}. The action of K_{i} on the basis element is to multiply by v^{m}, where m is the number of first coordinates in the basis element equal to i. The family of first coordinates is unchanged under the action of H, so the actions commute. The case of K_{i}^{−1} is similar, but with v^{−m} in place of v^{m}.

Now we consider the case of E_{i}. Assume firstly that b ≤ d. We wish to show that the actions of E_{i} and T_{s} commute. However, we know from the proof of Lemma 2.2.1 that the action of T_{s} on T^{r}(M_{n}), when projected to the first coordinates, is similar to the action of T_{s} on T^{r}(V), except that the two second coordinates are exchanged and the expression is multiplied by v^{−1} (if b < d) or v^{0} (if b = d), since we are working with unquantized basis elements. We know that the actions of E_{i} and T_{s} on tensor space commute. Because the action of U does not change second coordinates of matrix units, we see that the actions of E_{i} and T_{s} do indeed commute. Now assume that b > d. A similar argument now applies to the actions of E_{i} and T_{s}^{−1}. (Here, the second coordinates are exchanged by T_{s}^{−1} and the expression is multiplied by v.) Hence the actions of E_{i} and T_{s}^{−1} commute.

The same techniques as we used for E_{i} can be used to show that the action of F_{i} commutes with the action of H.

This completes the proof.

We are now ready to define the candidate for the monomorphism γ of algebras and U-modules from S_{v}(n, r) into T^{r}(M_{n}). Most of the rest of this chapter will be devoted to proving that γ satisfies the required properties.

Recall that S_{v}(n, r) has a basis denoted by {[A]}, as A ran over all n × n matrices with nonnegative integer entries summing to r. With each matrix A, we can (uniquely) associate a standard basis element t_{A} of T^{r}(M_{n}), given by

e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}},

exactly as in the classical case. Define D_{A} as in the classical case also. Define the integer m_{A} by

m_{A} := |{⟨i, j⟩ : 1 ≤ i < j ≤ r, b_{i} = b_{j}, a_{i} < a_{j}}|.

Example Let A be the matrix

( \begin{matrix}3 & 1 \\ 1 & 2 \end{matrix} ) .

Then t_{A} is given by

e_{11} ⊗ e_{11} ⊗ e_{11} ⊗ e_{21} ⊗ e_{12} ⊗ e_{22} ⊗ e_{22},

and m_{A} = 5.

Definition 2.2.3 The linear map γ : S_{v}(n, r) → T^{r}(M_{n}) is given by

γ([A]) := v^{−m_{A}}t_{A}. ( ∑_{w∈D_{A}} T_{w}) ,

where the action of H(A_{r−1}) on T^{r}(M_{n}) is as before.

Lemma 2.2.4

a) Let

t_{A} := e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}},

and let

t′_{A} := e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}}

be a standard basis element of T^{r}(M_{n}) such that d_{i} = b_{i} for all i and the ordered pairs ⟨c_{i}, d_{i}⟩ are a rearrangement of the ordered pairs ⟨a_{i}, b_{i}⟩. Then the coefficient of t′_{A} in γ([A]) is v^{−m′}, where

m′ := |{⟨i, j⟩ : 1 ≤ i < j ≤ r, d_{i} = d_{j}, c_{i} < c_{j}}|.

b) Any basis element t = e_{x_{1}b_{1}} ⊗ ⋯ ⊗ e_{x_{r}b_{r}} occurring with nonzero coefficient in γ([A]) must satisfy the hypotheses satisfied by t′_{A} above.

Proof We deal with the proof of a) first.

Firstly, we consider the action of ∑_{w∈D_{A}} T_{w} on [t_{A}], which of course differs from t_{A} by some power of v. Consider the action of a typical T_{w} in the sum, and write

w = s_{i_{1}} ⋯ s_{i_{m}}

as a reduced product of simple reflections.

We revert for a moment to acting S_{r} on T^{r}(M_{n}). Because w is a distinguished coset representative, we find that the matrix units occurring at positions i_{j} and i_{j+1} in

t_{j} := t_{A}.s_{i_{1}} ⋯ s_{i_{j−1}},

will be in the correct order with respect to the Hebrew lexicographic order (as defined in §1.4.1). (This follows from the fact that ℓ(s_{i_{1}} ⋯ s_{i_{j−1}}s_{i_{j}}) = ℓ(s_{i_{1}} ⋯ s_{i_{j−1}}) + 1, and the length identities quoted in §1.3.3.) In particular, if

t_{j} = v^{∗}e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}},

we have b_{i_{j}} ≤ b_{i_{j+1}}. (Here, “∗” denotes some integer whose value is unimportant.) Because of the above remark concerning the Hebrew lexicographic order, we deduce that when computing t_{A}.T_{w}, we never have to appeal to the case corresponding to b > d in the definition of the action of T_{s}.

This has repercussions for the quantized case. Recall the remarks in the proof of 2.2.1 about the similarities between the actions of T_{s} on tensor space and tensor matrix space, in the case b ≤ d— the action on the second coordinates is essentially permutation. Taking the coefficient of a typical quantized basis element appearing in [t_{A}].T_{w}, say

t := e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}},

we now know from the above discussion that if

t_{A} := e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}}

then the sequence (d_{1}, …, d_{r}) is simply the result of acting w on the right of the sequence (b_{1}, …, b_{r}) by place permutation.

Let W_{J} be the parabolic subgroup of W (A_{r−1}) which is generated by the (p, p + 1) satisfying b_{p} = b_{p+1}. We then see from the above discussion that unless w ∈ W_{J}, there will certainly be no nonzero coefficient of t′_{A} occurring in [t_{A}].T_{w} (because no basis elements occurring with nonzero coefficient in the latter expression can have their second coordinates in the correct order), and hence in t_{A}.T_{w}. This simplifies the problem considerably.

The next important observation is that, if a < b,

e_{ac} ⊗ e_{bc}.T_{s} = ve_{bc} ⊗ e_{ac}.

Informally, every time an element T_{s} (where s ∈ W_{J}) exchanges two matrix units out of Hebrew lexicographic order, the exponent of v is increased by 1. We know that there exists a unique element u ∈ W_{J} ∩ D_{A} such that t′_{A} = t_{A}.u (acting as an element of the symmetric group). (Conversely, for any element u′ ∈ W_{J} ∩D_{A}, there is a corresponding unique t′′_{A}, with properties similar to those of t′_{A}.) Write u as a reduced product of simple reflections, s_{i_{1}} ⋯ s_{i_{l}}. Then the entries in the i_{k} and i_{k+1}-th positions of t_{A}.s_{i_{1}} ⋯ s_{i_{k+1}} are in Hebrew lexicographic order with different first coordinates and equal second coordinates. Returning to the quantized case (which essentially differs only by powers of v), let the coefficient of t_{A} in γ([A]) be C. The number of pairs of matrix units exchanged out of order in passing from t_{A} to t′_{A} is m_{A} −m′, where m′ is as in the statement of the Lemma. Therefore, we find that the coefficient of t′_{A} in C.t_{A}. ∑_{w∈W_{J} ∩D_{A}} T_{w}, and hence in γ([A]), is C.v^{m_{A}−m′}, by using the above observations. However, we know from the definition of γ (and the correspondence between u′ and t′′_{A} mentioned above, in the case where u′ = 1 and t′′_{A} = t_{A}) that C = v^{−m_{A}}. Thus the coefficient of t′_{A} is v^{−m′} as claimed.

The proof of b) also follows from the preceding arguments. Let t = e_{x_{1}b_{1}} ⊗⋯⊗e_{x_{r}b_{r}}. We know that the only terms of γ([A]) which can yield this are of form t_{A}.T_{w}, where w ∈ W_{J} ∩ D_{A}. We have also shown that for such an element w, t_{A}.T_{w} is equal to

v^{∗}e_{y_{1}b_{1}} ⊗ ⋯ ⊗ e_{y_{r}b_{r}},

where the ⟨y_{i}, b_{i}⟩ are some rearrangement of the original ⟨a_{i}, b_{i}⟩. This completes the proof of b).

Before proceeding with the main theorem, it is worth defining the relaxed order, ⪯, on the matrix units. We say that e_{ab} ⪯ e_{cd} if (b < d) or (b = d and a > c) or (b = d and a = c). A basis element of T^{r}(M_{n}) is said to be relaxed if the matrix units yielding it appear in it in relaxed order. A basis element is called H-relaxed if the matrix units in it appear in Hebrew lexicographic order (for example, t_{A} for any basis matrix A). These definitions are compatible with the definitions in §1.4.1.

Example The basis element of T^{r}(M_{n}) given by

e_{21} ⊗ e_{11} ⊗ e_{11} ⊗ e_{11} ⊗ e_{22} ⊗ e_{22} ⊗ e_{12}

is relaxed.

One of the main points of Lemma 2.2.4 is that it gives the following result. Corollary 2.2.5

(i) In the expression γ([A]), with respect to the standard basis of T^{r}(M_{n}), the basis element R_{A} of T^{r}(M_{n}) which is the relaxed rearrangement of t_{A} is the only relaxed basis element appearing with nonzero coefficient. Furthermore, it appears with coefficient 1.

(ii) In the expression γ([A]), with respect to the standard basis of T^{r}(M_{n}), the basis element t_{A} is the only H-relaxed basis element appearing with nonzero coefficient. It appears with coefficient v^{−m_{A}}.

Proof Note that the element R_{A} is given by

\underbrace{e_{n1} ⊗ ⋯ ⊗ e_{n1}}_{A_{n1}} ⊗ ⋯ ⊗ \underbrace{e_{11} ⊗ ⋯ ⊗ e_{11}}_{A_{11}} ⊗ \underbrace{e_{n2} ⊗ ⋯ ⊗ e_{n2}}_{A_{n2}} ⊗ ⋯ ⊗ \underbrace{e_{12} ⊗ ⋯ ⊗ e_{12}}_{A_{12}} ⊗ · · · .

By definition, any relaxed basis element, R, appearing must have its second coordinates appearing in ascending order. Using Lemma 2.2.4, R must therefore be obtainable from t_{A} by action on the right by an element w ∈ W_{J} ∩ D_{A}, in the notation of the proof of 2.2.4. This means that R satisfies the hypotheses for t′_{A} in the statement of Lemma 2.2.4. Since R is relaxed, it must be equal to R_{A} (there is no alternative). We then find that the integer m′ defined in the Lemma is equal to 0, thus showing that R_{A} appears with coefficient 1.

The proof of (ii) uses the same techniques. In this case, w = 1 and m′ = m_{A}.

Example Let A be the matrix

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Then R_{A} = e_{21} ⊗ e_{12} and

γ([A]) = e_{12} ⊗ e_{21} + e_{21} ⊗ e_{12} + (v^{2} − 1)e_{22} ⊗ e_{11}.

Note that the coefficient of R_{A} in γ([A]) is 1.

We are now ready to show that γ is an isomorphism of U-modules. Lemma 2.2.6

(i) The images of the basis {[A]} of S_{v}(n, r) under γ span a U-submodule of T^{r}(M_{n}). (ii) The U-submodule defined above is also an H-submodule, and is the largest H-submodule of

T^{r}(M_{n}) on which T_{w} acts as the scalar q^{ℓ(w)}. (We will refer to this H-submodule as the ind submodule.)

(iii) The map γ is an isomorphism of U-modules between S_{v}(n, r) and the submodule described in

(i). Proof First, we compare γ([A]) with the element

g([A]) := t_{A}. ( ∑_{w∈W (A_{r−1})} T_{w}) .

Using the standard theory of Coxeter groups and Hecke algebras, and denoting by W_{A} the parabolic subgroup corresponding to D_{A}, we can rewrite g([A]) as

t_{A}. ( ∑_{u∈W_{A}} T_{u}) ( ∑_{w∈D_{A}} T_{w}) .

The point of doing this is that the action of T_{u}, where u ∈ W_{A}, on t_{A} is very simple—it simply multiplies by the scalar q^{ℓ(u)}. We know (see for example, [H, §3.15]) that the Poincaré polynomial of a Coxeter group of type A_{l} is

∏_{i=1}^{l} ∑_{k=0}^{i} t^{k},

and we also know that W_{A}, being a parabolic subgroup of a Coxeter group of type A, is canonically isomorphic to a direct product of such groups of smaller rank. Its Poincaré polynomial is therefore a product of expressions such as those given above. By substituting t = q, we obtain the scalar by which ∑_{u∈W_{A}} T_{u} acts, and since 1 + q + ⋯ + q^{e−1} ≠ 0 for any natural number e and ℚ(v) is a field, we find that this scalar is nonzero. Bringing the quantity v^{−m_{A}} back into the picture, we find that γ([A]) and g([A]) differ only by multiplication by a nonzero scalar. To prove (i), it is therefore enough to prove that the span of elements g([A]) is invariant under the action of U.

Before completing the proof of (i), it is convenient to prove (ii). Lemma 1.6 tells us that

( ∑_{w∈W} T_{w}) T_{d} = q^{ℓ(d)} ( ∑_{w∈W} T_{w}) .

This means that

g([A]).T_{d} = q^{ℓ(d)}g([A]),

by Lemma 2.2.1, and hence g([A]) lies in the ind submodule. We claim that any t ∈ T^{r}(M_{n}) satisfying

t.T_{w} = q^{ℓ(w)}t

for all w ∈ W (A_{r−1}) lies in the span of the elements g([A]). It is easy to see that

t = c.t ( ∑_{w∈W (A_{r−1})} T_{w})

for a certain scalar c ≠ 0. Let t_{1} = e_{a_{1}b_{1}} ⊗ ⋯ ⊗ e_{a_{r}b_{r}} be a basis element occurring in t. If we can prove that

t_{1}. ( ∑_{w∈W (A_{r−1})} T_{w})

lies in the span of the g([A]) then the proof of (ii) will follow.

We first deal with the case where b_{1} ≤ ⋯ ≤ b_{r}. The proof of Lemma 2.2.4 now shows that t_{1} = v^{∗}t_{A}.T_{w} for some H-relaxed basis element t_{A} and some w ∈ W (A_{r−1}). The assertion now follows by using Lemma 1.6. In the case where the sequence of b_{i} is not increasing, let (d_{1}, …, d_{r}) be the rearrangement of (b_{1}, …, b_{r}) satisfying d_{1} ≤ ⋯ ≤ d_{r}. Choose w such that

(d_{1}, …, d_{r}).w = (b_{1}, ⋯, b_{r}).

A simple inductive argument like the one in the proof of Lemma 2.2.4 shows that all elements occurring with nonzero coefficient in t_{1}.T_{w}^{−1} are of form

e_{x_{1}d_{1}} ⊗ ⋯ ⊗ e_{x_{r}d_{r}}.

We now use Lemma 1.6 and the previous case (where b_{1} ≤ ⋯ ≤ b_{r}) to deduce the required result, thus completing the proof of (ii).

We return to the proof of (i). Let u ∈ U. We know from Lemma 2.2.2 that

u. (t_{A}. ∑_{w∈W (A_{r−1})} T_{w}) = (u.t_{A}). ∑_{w∈W (A_{r−1})} T_{w},

and this latter element clearly lies in the ind submodule, and hence in the span of the elements g([A]). This completes the proof of (i).

For the proof of (iii), we compare the results we get by acting U on γ(S_{v}(n, r)) with those in [BLM]. It is enough to verify that the action of u on γ([A]) agrees with the results of [BLM] where u ∈ {E_{∗}, F_{∗}, K_{∗}, K_{∗}^{−1}} and [A] is a basis element for S_{v}(n, r). The results we are hoping for are: a)

K_{h}.γ([A]) = v^{s_{h}}γ([A])

and

K_{h}^{−1}.γ([A]) = v^{−s_{h}}γ([A]),

where s_{h} := ∑_{k=1}^{n} a_{hk}. b)

E_{h}.γ([A]) = ∑_{p∈[1,n]:a_{h+1,p}≥1} v^{β(p)}[a_{h,p} + 1] γ ([A + E_{h,p} − E_{h+1,p}]) ,

where

β(p) := ∑_{j>p}(a_{h,j} − a_{h+1,j}).

c)

F_{h}.γ([A]) = ∑_{p∈[1,n]:a_{h,p}≥1} v^{β′(p)}[a_{h+1,p} + 1] γ ([A + E_{h+1,p} − E_{h,p}]) ,

where

β′(p) := ∑_{j<p}(a_{h+1,j} − a_{h,j}).

Case a) is easy to verify, because

K_{h}.e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}} := v^{s}e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}},

where s is the number of c_{i} equal to h. By checking coefficients of relaxed terms, via Corollary 2.2.5, we see that the first half of a) holds. Similar techniques work for K_{h}^{−1}.

To prove case b), choose a basis element [B] which occurs in E_{h}.[A]. It is enough to check that the coefficient of [B] is correct. Because of the way E_{h} acts on T^{r}(M_{n}), we can find a close relationship between the relaxed basis elements R_{B} and R_{A}. By using Lemma 2.2.4 b), we see that the only difference is that the rightmost occurrence of e_{h+1,p} in R_{A} is replaced by e_{h,p} to obtain R_{B}, for some integer p : 1 ≤ p ≤ n. (If this cannot happen, then the coefficient must be zero as desired.) Let us say that this difference occurs between the P -th component of the basis elements.

We need to identify all basis elements occurring with nonzero coefficients in γ([A]) whose images under the action of E_{h} contain a nonzero coefficient of R_{B}. We do this by considering the action of E_{h} on T^{r}(M_{n}). An example of such an element is R_{A} itself. A typical basis element R satisfying these hypotheses differs from R_{A} just to the right of the P -th component; where R_{A} contains the sequence

⋯ ⊗ e_{h+1,p} ⊗ e_{h,p} ⊗ ⋯ ⊗ e_{h,p} ⊗ ⋯,

R contains some permutation of this sequence. (Informally, e_{h+1,p} may occur a few places to the right of where it “should”.) Denote by R_{z} the basis element R (as above) where the rightmost occurrence of e_{h+1,p} occurs at place P + z (so in particular, R_{0} = R_{A}). Let h(z) denote the multiplicity of e_{h,p} in R_{A}. (It is equal to A_{h,p}.)

We know from Lemma 2.2.4 that R_{z} occurs in γ([A]) with coefficient v^{−z}.

It can be shown that

Δ^{r−1}(E_{h}) = 1 ⊗ ⋯ ⊗ E_{h} + 1 ⊗ ⋯ ⊗ K_{i}K_{i+1}^{−1} ⊗ E_{h} + ⋯ + E_{h} ⊗ K_{i}K_{i+1}^{−1} ⊗ ⋯ ⊗ K_{i}K_{i+1}^{−1},

where there are r tensor components in each summand. Using this fact, we find that the contribution of v^{−z}R_{z} to the coefficient of R_{B} in the result is v^{x_{z}}, where

x_{z} := −z + h(z) − z + ( ∑_{p′>p} A_{h,p′} ) − ( ∑_{p′>p} A_{h+1,p′} ) .

Summing over all possible z (i.e. 0 ≤ z ≤ h(z)), we find that the total contribution from γ([A]) to R_{B} is v^{β(p)}[A_{h,p} + 1], as in the statement of the Lemma.

Case c) is very similar to case b), so we omit the proof.

This completes the proof of the Lemma.

Now that we know that γ is an isomorphism of U-modules, it is not too hard to show that it is an isomorphism of algebras. We know that S_{v}(n, r) is generated as an algebra by

θ(E_{h}), θ(F_{h}), θ(K_{i}), θ(K_{i}^{−1}),

where 1 ≤ h < n and 1 ≤ i ≤ n. It therefore suffices to show that

γ(θ(u) × [A]) = γ(θ(u)) × γ([A]),

for all [A] and for all u in the standard algebra generating set {E_{∗}, F_{∗}, K_{∗}, K_{∗}^{−1}} of U.

Theorem 2.2.7 The map γ is a canonical monomorphism of algebras and of U-modules from S_{v}(n, r) to T^{r}(M_{n}). Proof Recall the remark after Proposition 2.1.3. Again, it is true that the action of u ∈ U on an element t ∈ T^{r}(M_{n}) is given by

u.t = γ_{r}(u) × t,

using the quantized version of the map γ_{r} appearing in Proposition 2.1.3. We claim that it is enough to show that γ_{r}(u) = γ(θ(u)) for elements u in the standard generating set. For if this is true, Lemma 2.2.6, the preceding remark, and the fact that

u.[A] := θ(u) × [A]

show that

γ(θ(u) × [A]) = γ(u.[A]) = u.γ([A]) = γ_{r}(u) × γ([A]) = γ(θ(u)) × γ([A]),

as required.

The case u = K_{i} is fairly easy, so we tackle this first. Beilinson et al. [BLM] define

θ(K_{i}) = ∑_{D∈𝔻_{r}} v^{d_{i}}[D],

where 𝔻_{r} is the set of n × n diagonal matrices with nonnegative integer entries summing to r. Calculating γ([D]), where D is a diagonal matrix, turns out to be remarkably simple, because m_{D} = 0 and because the Hecke algebra H(A_{r−1}) acts via permutations! The reason for the latter is as follows. Consider a typical element T_{w} ∈ D_{D}, and write w = s_{i_{1}} ⋯ s_{i_{l}} as a reduced expression, as usual. Let

t_{j} := t_{D}.s_{i_{1}} ⋯ s_{i_{j−1}}.

As in the proof of Lemma 2.2.4, the matrix units occurring at positions i_{j} and i_{j+1} in t_{j} are lexicographically ordered. However, if a < b,

e_{aa} ⊗ e_{bb}.T_{s} = e_{bb} ⊗ e_{aa},

so we see that H acts only via permutations. From this, we see that

γ(θ(K_{i})) = ∑_{D} (v^{d_{i}}.t_{D}. ( ∑_{w∈D_{D}} T_{w})) ,

where the outer sum ranges over all possible values of the diagonal matrices D = diag(d_{1}, …, d_{r}) subject to the constraint ∑ d_{i} = r.

We now check that this is equal to γ_{r}(K_{i}). It is not hard to show that

Δ^{r−1}(K_{i}) = \underbrace{K_{i} ⊗ ⋯ ⊗ K_{i}}_{r times} .

We know that

θ_{1}(K_{i}) = ∑_{j=1}^{n} v^{δ(i,j)}e_{j,j}.

Using the fact that

γ_{r} := \underbrace{(θ_{1} ⊗ ⋯ ⊗ θ_{1})}_{r times} Δ^{r−1},

it is fairly easy to see that the maps γ_{r} and γθ agree on K_{i}, as required.

The case u = K_{i}^{−1} follows by an argument entirely analogous to the previous one.

We now tackle the case of u = E_{h}. Following [BLM, §5.7], we have

θ(E_{h}) := ∑_{D∈𝔻_{r−1}}[E_{h,h+1} + D].

We then use a similar technique to the one used in the K_{i} case to argue that H “almost” acts via permutations (more precisely—it does act via permutations, but introduces certain powers of v). The situation is very similar to the K_{i} situation, except that

e_{hh+1} ⊗ e_{h+1h+1}.T_{s} = ve_{h+1h+1} ⊗ e_{hh+1}

and

e_{hh} ⊗ e_{hh+1}.T_{s} = ve_{hh+1} ⊗ e_{hh}.

Informally, this means that the situation is similar to the classical case, except the processes of passing an e_{h+1h+1} to the left of the e_{hh+1} and that of passing an e_{hh} to the right of an e_{hh+1} multiply the relevant basis element by v. However, we know from Corollary 2.2.5 that “relaxed” basis elements have coefficient 1. Such basis elements contain e_{h,h+1} to the right of any possible occurrences of e_{hh} and e_{h+1h+1}. This means that the coefficient of any typical basis element of T^{r}(M_{n}) occurring in γ(θ(E_{h})), say

t = e_{c_{1}d_{1}} ⊗ ⋯ ⊗ e_{c_{r}d_{r}},

occurs with coefficient v^{s}, where s is the number of e_{hh} occurring to the right of the occurrence of e_{hh+1} minus the number of e_{h+1h+1} occurring to the right of the occurrence of e_{hh+1}.

We know that

Δ^{r−1}(E_{h}) = 1 ⊗ ⋯ ⊗ E_{h} + 1 ⊗ ⋯ ⊗ K_{i}K_{i+1}^{−1} ⊗ E_{h} + ⋯ + E_{h} ⊗ K_{i}K_{i+1}^{−1} ⊗ ⋯ ⊗ K_{i}K_{i+1}^{−1},

where there are r tensor components in each summand. Since θ_{1}(E_{h}) = e_{hh+1} and we know what the θ_{1}(K_{i}^{±1}) are, it is possible to see from the above discussion that the maps γ_{r} and γθ agree on E_{h}.

The case of u = F_{h} is entirely analogous to that of u = E_{h}.

This completes the proof of the main result.

Example Consider S_{v}(2, 2). Let

A := ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

and let I be the 2 × 2 identity matrix. Let e_{A} and e_{I} be the corresponding elements in the Dipper-James basis of S_{v}(2, 2). We know from properties of the Hecke algebra of type A_{1} embedded in S_{v}(2, 2) that e_{A}^{2} = qe_{I} + (q − 1)e_{A}, but we wish to show this directly. It can be checked that v[A] = e_{A} and [I] = e_{I}.

Calculating the values of γ on these basis elements, we find that

v.γ([A]) = ve_{12} ⊗ e_{21} + ve_{21} ⊗ e_{12} + (v^{2} − 1)e_{22} ⊗ e_{11},

and that

γ([I]) = e_{11} ⊗ e_{22} + e_{22} ⊗ e_{11}.

Squaring v.γ([A]) in the ring T^{r}(M_{n}), we find that it gives

v^{2}e_{11} ⊗ e_{22} + v^{2}e_{22} ⊗ e_{11} + (v^{2} − 1)^{2}e_{22} ⊗ e_{11} + (v^{3} − v)e_{21} ⊗ e_{12} + (v^{3} − v)e_{12} ⊗ e_{21}.

The relaxed basis elements occurring in these expressions are e_{11} ⊗ e_{22}, corresponding to [I], and occurring with coefficient q = v^{2}, and e_{21} ⊗ e_{12}, corresponding to [A] and occurring with coefficient (v^{3} − v) = (q − 1)v. Hence we have

(v[A])^{2} = q[I] + (q − 1)(v[A]),

thus proving the desired result.

Remarks on Theorem 2.2.7 As can be seen from the example, these methods give a fairly easy way of computing products in S_{v}(n, r), without having to resort to difficult procedures involving flags and finite fields.

Since the classical case corresponding to this (i.e. when q and v are replaced by 1) gives the situation where the classical Schur algebra is the r-th symmetric power of the n × n matrix algebra, the Theorem casts more light on the nature of the quantization. The space T^{r}(M_{n}) provides an arena in which the quantization takes place.

It should be noted that this product rule also gives a product rule for S_{q}(n, r) in terms of the more usual basis {e_{A}}. In [BLM, Lemma 2.3], it is proved that e_{A} = v^{s_{A}}[A], where

s_{A} := ∑_{1≤i,j,k,l≤n} A_{i,j}A_{k,l},

where the sum is taken with the restrictions that i ≥ k and j < l. This exhibits a close connection between the two bases. The restrictions on the sum are reminiscent of the relaxed order, ⪯, defined on the matrix units.

The product rule we obtain from using the above Theorem is of a convenient form for proving theorems whose classical analogues are proved using Schur’s Product Rule.

2.3. On certain subalgebras of the q-Schur algebra

Using the description of the v-Schur algebra given in §2.2, we can describe a (fairly large) family of subalgebras of the q-Schur algebra which are spanned by certain natural basis elements e_{A}. To do this, we introduce the notion of a “good” set of matrix units.

Definition A subset G of {e_{ij} : 1 ≤ i, j ≤ n} is called a good set of matrix units if it satisfies the following two conditions:

(i) G spans a subalgebra of the full matrix ring M_{n}; (ii) if e_{ab} ∈ G and e_{cd} ∈ G where a < c and b > d, then also e_{ad} ∈ G and e_{cb} ∈ G.

Lemma 2.3.1 Let G be a good set of matrix units. Then T^{r}(G) is a subalgebra of T^{r}(M_{n}), and a right H-submodule of T^{r}(M_{n}). Proof It is clear that T^{r}(G) is a subalgebra of T^{r}(M_{n}) because the elements of G span a subalgebra of M_{n}.

We now need to check that if [e_{i_{1}j_{1}} ⊗⋯⊗e_{i_{r}j_{r}}] ∈ T^{r}(G), then acting T_{s} (where s = (p, p+1) for some p) on this element produces another element of T^{r}(G). Because of the way the multiplication is defined, we need only check this in the case r = 2.

We now consider the matrix units appearing in the expression for [e_{ab} ⊗e_{cd}].T_{s} (where s = (12)) as in the definition of the action of H on T^{r}(M_{n}). It is easily checked that if a ≤ c and b ≤ d, then no new matrix units can appear in the result. It is similarly easily checked that if a > c and b > d, no new matrix units can appear in the result. If a > c and b = d, or if a = c and b > d, similar remarks hold.

We are left with the two most interesting cases, namely the case when a > c and b < d, and the case when a < c and b > d. In both cases, we find that e_{ad} and e_{cb} are also elements of G, from the definition of a good set of matrix units. It is now clear from the definition of the action of H that all matrix units appearing in the result are elements of G, as desired.

Definition If G is a good set of matrix units, we denote by S_{v}(G) the subspace of S_{v}(n, r) spanned by all basis elements [A] where A is a matrix such that A_{ij} ≠ 0 ⇒ e_{ij} ∈ G. We denote by S_{q}(G) the corresponding subspace of S_{q}(n, r) spanned by elements e_{A}.

Lemma 2.3.2 Using the monomorphism γ defined in §2.2, the following identity holds in T^{r}(M_{n}):

γ(S_{v}(n, r)) ∩ T^{r}(G) = γ(S_{v}(G)).

Proof We first establish that γ(S_{v}(G)) ≤ γ(S_{v}(n, r)) ∩ T^{r}(G). Clearly S_{v}(G) ≤ S_{v}(n, r). It is now enough to show that γ([A]) ∈ T^{r}(G) where [A] is a basis element of S_{v}(G). By definition of γ, γ([A]) is of the form t_{A}.h for some h ∈ H. By the construction of [A], we find that t_{A} ∈ T^{r}(G), and therefore γ([A]) ∈ T^{r}(G) by Lemma 2.3.1.

Now consider a typical element in γ(S_{v}(n, r)) ∩ T^{r}(G) given by

g = ∑_{A} c_{A}γ([A]),

where the c_{A} are elements of ℚ(v). We claim that unless [A] is a basis element of S_{v}(G), c_{A} must be zero. This is achieved by considering relaxed basis elements occurring in the expression for g. These relaxed basis elements must all lie in T^{r}(G). It follows from the definitions that the relaxed basis element corresponding to [A] is an element of T^{r}(G) if and only if [A] is a basis element for S_{v}(G). This establishes the reverse inclusion.

Theorem 2.3.3 S_{v}(G) is a subalgebra of S_{v}(n, r) and S_{q}(G) is a subalgebra of S_{q}(n, r). The dimension of each subalgebra is

\binom{m + r − 1}{r},

where m is the cardinality of the set G. Proof We know that S_{v}(G) is a subspace of S_{v}(n, r) and it is clearly a subalgebra by Lemma 2.3.2, where it was shown to be an intersection of two subalgebras of T^{r}(M_{n}). The corresponding result for S_{q}(G) follows immediately by consideration of products e_{A}e_{A′} where both elements are in S_{q}(G).

By construction of the basis for S_{v}(G) or S_{q}(G), we see that the dimension is equal to the number of compositions of r into m pieces. The result now follows from elementary combinatorics.

Examples

We now present some examples of good sets of matrix units.

1. Let

G = {e_{ij} : 1 ≤ i ≤ j ≤ n}.

This spans the subalgebra of M_{n} given by upper triangular matrices. It is easily checked that G is good, and that S_{q}(G) is the Borel subalgebra S^{+}.

2. Choose a finite sequence of integers r_{i} satisfying 0 = r_{0} < r_{1} < … < r_{s} = n. Then let

G = {e_{ij} : 1 ≤ i, j ≤ n and (r_{h} < i ≤ r_{h+1} ⟺ r_{h} < j ≤ r_{h+1})}.

This spans the subalgebra of M_{n} given by matrices of a particular block form (depending on the sequence r_{i}). The set G is good, and the algebra S_{q}(G) ≤ S_{q}(n, r) exists.

3. Let

G = {e_{11}} ∪ {e_{ij} : 2 ≤ i ≤ n, 1 ≤ j ≤ n}.

This spans a subalgebra of M_{n}. Furthermore, G is good and the algebra S_{q}(G) ≤ S_{q}(n, r) exists.

3. A Straightening Formula for

Quantized Codeterminants

In this chapter, we introduce q-analogues of codeterminants, which were defined in the classical case by J.A. Green [G2, §4]. These were shown to be parametrised by pairs of row-semistandard tableaux of shape λ, where λ ∈ Λ(n, r). If both the tableaux are standard, and the composition λ is dominant, then the codeterminant is said to be standard. The set of standard codeterminants forms a free ℤ-basis for the integral form of the classical Schur algebra. An algorithm for expressing an arbitrary codeterminant as a sum of standard ones has been described in [W].

The main results of this chapter are Theorem 3.2.6 and Corollary 3.2.7, which prove constructively that the set of standard q-codeterminants forms a free ℤ[q, q^{−1}]-basis of the q-Schur algebra, S_{q}(n, r). This fact has several important applications later in the thesis, and the main results in §4, §5 and §6 rely on it fairly heavily.

3.0. Conventions and Review of §2

We will write v^{∗} to mean some integer power of v. Usually no attempt will be made to keep track of the value.

In some of the proofs in this chapter, we will use the technique of proving that an element w ∈ W, where W is a Coxeter group, satisfies a certain property P by “induction on left subwords of w.” The idea is as follows. Let s_{i_{1}} ⋯ s_{i_{m}} be any fixed reduced expression for w. We then prove by induction on p (where 1 ≤ p ≤ m) that the element w_{p} defined by s_{i_{1}} ⋯ s_{i_{p}} satisfies the property P; this then proves that w has the required property. It is immediate from the properties of Coxeter groups that if w is a distinguished right coset representative with respect to some parabolic subgroup of W, then so are all its left subwords. It is also immediate that if w lies in some parabolic subgroup W_{P} of W, then so do all its left subwords.

It is convenient to recall the action of H on the right of T^{r}(M_{n}) given in §2.2, but this time with respect to a slightly different basis.

We now recall the quantized basis of T^{r}(M_{n}). This was introduced in §2.2. We now re-express this as the set {[u_{i,j}] : i, j ∈ I(n, r)}. Here, [u_{i,j}] is defined to be v^{−m(i)−m(j)}u_{i,j}, where the function m is as in §1.3.6.

It was shown in Lemma 2.2.1 that T^{r}(M_{n}) admits a right H-module structure which we now recall. Let T_{s} = T_{(p,p+1)}. Then:

[u_{i,j}].T_{s} = \{ \begin{matrix}q[u_{i.s,j.s}] & if i_{p} ≤ i_{p+1} and j_{p} ≤ j_{p+1};  (1) \\ [u_{i.s,j.s}] + (q − 1)[u_{i,j}] & if i_{p} > i_{p+1} and j_{p} > j_{p+1};  (2) \\ [u_{i.s,j.s}] + (q − 1)[u_{i,j.s}] & if i_{p} > i_{p+1} and j_{p} ≤ j_{p+1};  (3) \\ q[u_{i.s,j.s}] + (q − 1)[u_{i,j}] − (q − 1)[u_{i,j.s}] & if i_{p} ≤ i_{p+1} and j_{p} > j_{p+1}.  (4) \end{matrix}

From this, the action of T_{(p,p+1)}^{−1} immediately follows. It is given by:

[u_{i,j}].T_{s}^{−1} = \{ \begin{matrix}q^{−1}[u_{i.s,j.s}] & if i_{p} ≥ i_{p+1} and j_{p} ≥ j_{p+1};  (5) \\ [u_{i.s,j.s}] + (q^{−1} − 1)[u_{i,j}] & if i_{p} < i_{p+1} and j_{p} < j_{p+1};  (6) \\ [u_{i.s,j.s}] + (q^{−1} − 1)[u_{i,j.s}] & if i_{p} < i_{p+1} and j_{p} ≥ j_{p+1};  (7) \\ q^{−1}[u_{i.s,j.s}] + (q^{−1} − 1)[u_{i,j}] − (q^{−1} − 1)[u_{i,j.s}] & if i_{p} ≥ i_{p+1} and j_{p} < j_{p+1}.  (8) \end{matrix}

Note 3.0.1

Notice that this action specialises (when q and v are replaced by 1) to permutation, and that any basis element occurring with nonzero coefficient in u_{a,b}.T_{w} is of form u_{c,d} where a ∼ c and b ∼ d.

The rest of this chapter falls into two main sections. In §3.1, we show that any q-codeterminant can be expressed as a ℤ[q, q^{−1}]-linear sum of codeterminants of dominant shape. In §3.2, we prove a quantum analogue of the straightening formula for codeterminants. This is a procedure for expressing any codeterminant of dominant shape as a ℤ[q, q^{−1}]-linear sum of standard codeterminants. This was proved by Woodcock [W] in the classical case. We use some of the techniques in that paper to prove the quantum analogue, but some steps are considerably more delicate and require new methods.

3.1. Hecke algebras and codeterminants of dominant shape Lemma 3.1.1 Let t = u_{a,b} (where a, b ∈ I(n, r)) be a basis element of T^{r}(M_{n}) of parabolic form. Then

y := t.\frac{x_{W}}{P_{t}}

lies in S_{v}(n, r). Moreover, if t is expressed as an A-linear sum of elements ξ_{i,j}, then all elements occurring with nonzero coefficients in the sum satisfy i ∼ a and j ∼ b. Proof Lemma 1.6 shows that

T_{d}x_{W} = q^{ℓ(d)}x_{W} ,

where W is the Coxeter group of type A_{r−1}. We now find that y lies in the ind submodule of T^{r}(M_{n}), and hence in S_{v}(n, r) by Lemma 2.2.6. This proves that t is a ℚ(v)-linear combination of elements ξ_{i,j}. Thanks to Corollary 2.2.5, it is now enough to show that the coefficients of the relaxed basis elements occurring in y lie in A, because that will force the coefficient of each ξ_{i,j} to lie in A.

Define W_{J} to be the parabolic subgroup of W generated by the (p, p + 1) satisfying b_{p} = b_{p+1}. Let t_{0} be the unique element of form u_{a′,b} satisfying (a′, b) ∼ (a, b) and the condition that if b_{p} = b_{p+1} then a′_{p} ≤ a′_{p+1}. Let W_{I} be the parabolic subgroup of S_{r} stabilising t_{0}. There exists a unique w ∈ D_{I} ∩ W_{J} such that a′.w = a. Then we see by induction on left subwords of w that u_{a′,b}.T_{w} = v^{∗}u_{a,b}, as required. By substituting t_{0} for t in the definition of y, and by using the observation that if a ∈ A then v^{m}a ∈ A for any m ∈ ℤ, we find that it is enough to prove the lemma in the case where t = t_{0}, because P_{t} = P_{t_{0}}.

We now assume t = t_{0} and define the parabolic subgroup W_{I} to be generated by all (p, p + 1) such that a′_{p} = a′_{p+1} and b_{p} = b_{p+1}. Let D_{I} be the set of distinguished right coset representatives corresponding to W_{I}. Then, using the theory of Coxeter groups, we have

x_{W} = x_{I} ( ∑_{u∈D_{I}} T_{u}) .

We know (see for example, [H, §3.15]) that the Poincaré polynomial of a Coxeter group of type A_{l} is

∏_{i=1}^{l} ∑_{k=0}^{i} z^{k},

and we also know that W_{I}, being a parabolic subgroup of a Coxeter group of type A, is canonically isomorphic to a direct product of such groups of smaller rank. Its Poincaré polynomial is therefore a product of expressions such as those given above. By substituting z = q, we obtain the scalar by which ∑_{u∈W_{I}} T_{u} acts, and we find that

t.x_{I} = P_{t}.t.

This means that

y = t. ( ∑_{u∈D_{I}} T_{u}) ,

and hence, since the action of H on T^{r}(M_{n}) is defined over A, that the coefficients of any basis elements occurring in y, in particular the relaxed ones, lie in A.

The last assertion follows from Corollary 2.2.5 and Note 3.0.1.

Definition Let i, j ∈ I(n, r) be given by i = (i_{1}, …, i_{r}) and j = (j_{1}, …, j_{r}). Then the element τ_{i,j} ∈ T^{r}(M_{n}) is defined as

τ_{i,j} := u_{i,j}.\frac{x_{W}}{P_{i,j}} .

If j is of parabolic form, we know from Lemma 3.1.1 that τ_{i,j} can be considered to lie in S_{v}(n, r).

Note that if the u_{i,j} constructed above is H-relaxed, then τ_{i,j} is equal to ξ_{i,j} multiplied by a certain power of v.

Also note that Lemma 3.1.1 tells us that each element τ_{i,j} is of form

∑ c_{i′,j′}ξ_{i′,j′},

where the elements c_{i′,j′} lie in A, and the indices i′ and j′ satisfy i′ ∼ i and j′ ∼ j.

During the proofs that follow, we shall need a way of testing whether two elements of H are equal to each other. To do this, we construct a right H-module M with the property that there exist certain elements m ∈ M such that if h_{1}, h_{2} ∈ H and m.h_{1} = m.h_{2} then h_{1} = h_{2}. By linearity, it is enough to show that if m.h = 0 then h = 0. Lemma 3.1.2 Let V be a ℚ(v)-vector space of dimension n with basis e_{1}, …, e_{n}. We make M := V^{⊗r} into a right H-module in the usual way (i.e. via the tensor space action defined in §1.3.6). Let h ∈ H. Then:

(i) if m := e_{1} ⊗ ⋯ ⊗ e_{r}, then m.h = 0 ⇒ h = 0; (ii) if m_{π} := e_{π(1)} ⊗ ⋯ ⊗ e_{π(r)}, where π ∈ S_{r}, then m_{π}.h = 0 ⇒ h = 0. Proof We first prove (i). Let m and m_{π} be defined as in the statement of the Lemma. By using induction on ℓ(π), we find that m.T_{π} = v^{ℓ(π)}m_{π}. If π ≠ π′ then m_{π} ≠ m_{π′}; in fact, the set {m_{π} : π ∈ S_{r}} is linearly independent in V^{⊗r}. However, {T_{w} : w ∈ S_{r}} is known to be a basis for H. Since v is invertible, we now find that if m.h = 0, then h = 0.

We now prove (ii). We know from the above argument that m.T_{π} = v^{ℓ(π)}m_{π}. Assuming that m_{π}.h = 0, then m.(v^{−ℓ(π)}T_{π}h) = 0. Now (i) shows that v^{−ℓ(π)}T_{π}h = 0, and, since v and T_{π} are both invertible, that h = 0. This completes the proof.

We now need to introduce a family of bases for H, of which the usual basis {T_{w}} is an example. The motivation behind doing this is that by picking the correct basis, it is possible to deduce certain results concerning the coefficients of basis elements in the τ_{i,j} which were introduced earlier. This will be important later.

For convenience, we now choose, once and for all, a “favourite” reduced expression ρ(w) for each element w ∈ W. Definition Let j ∈ I(n, r) be of parabolic form, let W_{J} be its associated parabolic subgroup, and let D_{J} be the associated set of distinguished right coset representatives. To each element w ∈ D_{J}, we will associate an element D_{w}^{j} ∈ H which specialises as q is replaced by 1 to w in the group algebra.

Suppose ρ(w) = s_{i_{1}} ⋯ s_{i_{m}}. Earlier, we showed how S_{r} acts on I(n, r) on the right. For each integer s such that 1 ≤ s ≤ m we define an element h_{s} of H as follows. Consider j′ := j.s_{i_{1}} ⋯ s_{i_{s−1}}. Because w is a distinguished right coset representative, it must be the case that j′_{i_{s}} ≠ j′_{i_{s+1}}. (This follows from the length identities in §1.3.3, the definition of distinguished right coset representatives, and the fact that any left segment of a distinguished right coset representative is another distinguished right coset representative.) If j′_{i_{s}} > j′_{i_{s+1}} then h_{s} := T_{i_{s}}^{−1}, otherwise h_{s} := T_{i_{s}}. We then define D_{w}^{j} to be h_{1}h_{2} ⋯ h_{m}.

Remark

If 1 ≤ a < b ≤ n, then e_{b} ⊗ e_{a}.T_{s}^{−1} = v^{−1}e_{a} ⊗ e_{b} and e_{a} ⊗ e_{b}.T_{s} = ve_{b} ⊗ e_{a}. Define, for any i ∈ I(n, r),

m(i) := |{⟨a, b⟩ : 1 ≤ a < b ≤ r, i_{a} > i_{b}}|.

(For example, if i = (π(1), …, π(r)) as in the statement of Lemma 3.1.2 (ii), then m(i) = ℓ(π).)

If s = (p, p + 1) and j_{p} ≤ j_{p+1} then u_{i,j}.T_{s} is a linear combination of elements u_{i′,j.s} (see §3.0,

(1) and (3)). If j_{p} ≥ j_{p+1} then u_{i,j}.T_{s}^{−1} is a linear combination of elements u_{i′,j.s} (see §3.0, (5) and (7)).

The crucial property of the element D_{w}^{j} is as follows:

v^{m(j)}e_{j_{1}} ⊗ ⋯ ⊗ e_{j_{r}}.D_{w}^{j} = v^{m(j.w)}e_{w(j_{1})} ⊗ ⋯ ⊗ e_{w(j_{r})}.

This can be seen by induction on left subwords of w.

It is also follows that basis elements occurring with nonzero coefficients in u_{i,j}.D_{w}^{j} are of form u_{i′,j.w}.

Example

Let j = (4, 4, 1, 2). Then the parabolic subgroup W_{J} is generated by s_{1} = (1, 2). A routine check shows that the element w ∈ S_{4} given by

( \begin{matrix}1 & 2 & 3 & 4 \\ 2 & 4 & 3 & 1 \end{matrix} )

lies in D_{J}. The element j.w ∈ I is equal to (2, 4, 1, 4). A reduced expression ρ(w) for w is given by s_{3}s_{2}s_{1}s_{3}. The element D_{w}^{j} ∈ H is now given by T_{3}T_{2}^{−1}T_{1}^{−1}T_{3}^{−1}. (It will be seen later that the element D_{w}^{j} is independent of the reduced expression chosen for w.)

We now compare the elements D_{w}^{j} with their counterparts T_{w}. To do this, it is convenient to define an element R(j) ∈ I(r, r) (N.B.—not necessarily in I(n, r)) which is in some sense a “refinement” of j. We will then act various D_{w}^{j} on this R(j), and appeal to Lemma 3.1.2 to relate them to the usual basis consisting of the T_{w}. Definition Let i ∈ I(n, r) be of parabolic form. Then the element R(i) is the unique element of I(r, r) satisfying the following properties:

(i) R(i) = (π(1), …, π(r)) for some π ∈ S_{r}; (ii) if 1 ≤ a < b ≤ r and i_{a} < i_{b} (respectively i_{a} > i_{b}) then R(i)_{a} < R(i)_{b} (respectively R(i)_{a} >

R(i)_{b}); (iii) if 1 ≤ a < b ≤ r and i_{a} = i_{b} then R(i)_{a} < R(i)_{b}. Example Let n = 4, r = 8 and i = (1, 1, 1, 4, 4, 4, 3, 3). Then R(i) = (1, 2, 3, 6, 7, 8, 4, 5). Remarks Note that m(R(i)) = m(i) for any i ∈ I(n, r) of parabolic form. Also note that

v^{m(R(i))}e_{R(i)_{1}} ⊗ ⋯ ⊗ e_{R(i)_{r}}.D_{w}^{i} = v^{m(R(i).w)}e_{R(i)_{w(1)}} ⊗ ⋯ ⊗ e_{R(i)_{w(r)}}.

Together with Lemma 3.1.2, this shows that the element D_{w}^{j} is in fact independent of the reduced expression we were using for w.

We can say more than this: if W_{I} is the parabolic subgroup associated with i, and u ∈ W_{I}, then

v^{m(R(i))}e_{R(i)_{1}} ⊗ ⋯ ⊗ e_{R(i)_{r}}.T_{u}D_{w}^{i} = v^{m(R(i).uw)}e_{R(i)_{uw(1)}} ⊗ ⋯ ⊗ e_{R(i)_{uw(r)}}.

This follows from the definitions of R(i) and D_{w}^{i}: all the actions are essentially permutations, but with powers of v introduced to keep track of the changing value of the m function (which can be thought of as “length”).

Lemma 3.1.3 Let j ∈ I(n, r) be of parabolic form, with associated parabolic subgroup W_{J}. Then: (i)

( ∑_{u∈W_{J}} T_{u}) ( ∑_{w∈D_{J}} D_{w}^{j}) = q^{−m(j)} ∑_{w∈W} T_{w}.

(ii) The set

{T_{u}D_{w}^{j} : u ∈ W_{J}, w ∈ D_{J}}

is a basis for H.

Proof Let π ∈ S_{r} be such that R(j) = (π(1), …, π(r)). Then

e_{1} ⊗ ⋯ ⊗ e_{r}.T_{π} = v^{m(R(j))}e_{R(j)_{1}} ⊗ ⋯ ⊗ e_{R(j)_{r}}.

It now follows that

v^{m(R(j))}e_{R(j)_{1}} ⊗ ⋯ ⊗ e_{R(j)_{r}}.T_{π}^{−1}. ( ∑_{w∈W} T_{w}) = ∑_{w∈W} v^{ℓ(w)}e_{w(1)} ⊗ ⋯ ⊗ e_{w(r)}.

It also follows from the preceding remarks that

v^{m(R(j))}e_{R(j)_{1}} ⊗ ⋯ ⊗ e_{R(j)_{r}}. ( ∑_{u∈W_{J}} T_{u}) ( ∑_{w∈D_{J}} D_{w}^{j}) = ∑_{w∈W} v^{ℓ(w)}e_{w(1)} ⊗ ⋯ ⊗ e_{w(r)}.

Applying Lemma 3.1.2 (ii), this shows that

( ∑_{u∈W_{J}} T_{u}) ( ∑_{w∈D_{J}} D_{w}^{j}) = T_{π}^{−1}. ( ∑_{w∈W} T_{w}) ,

where the right-hand side is known to be equal to

q^{−ℓ(π)} ∑_{w∈W} T_{w},

thus establishing (i) since ℓ(π) = m(R(j)) = m(j).

For the proof of (ii), note that the set

{v^{m(R(j))}e_{R(j)_{1}} ⊗ ⋯ ⊗ e_{R(j)_{r}}.T_{u}D_{w}^{j} : u ∈ W_{J}, w ∈ D_{J}}

is linearly independent in T^{r}(V), because each element is a power of v times a certain basis element, and basis elements corresponding to distinct elements are nontrivial permutations of each other. Lemma 3.1.2 (ii) now shows that the set

{T_{u}D_{w}^{j} : u ∈ W_{J}, w ∈ D_{J}}

is independent in H, and, since it has the correct cardinality, that it is a basis for H.

Definition Let z ∈ I(n, r). Define z^{+} to be the element of I(n, r) satisfying z_{i}^{+} = f(z_{i}) for all i and for f ∈ S_{n} of minimal length subject to the condition that wt(z^{+}) ∈ Λ^{+}(n, r). (Such an f is uniquely defined, as will be illustrated in the following example.) Example Let n = 5, r = 9 and z = (1, 1, 2, 2, 2, 2, 4, 4, 4), so that wt(z) = (2, 4, 0, 3, 0). Then the permutation

( \begin{matrix}1 & 2 & 3 & 4 & 5 \\ 3 & 1 & 4 & 2 & 5 \end{matrix} )

works in the above definition of f, and z^{+} = (3, 3, 1, 1, 1, 1, 2, 2, 2), with wt(z^{+}) = (4, 3, 2, 0, 0). Note that wt(z) is a permutation of wt(z^{+}), and that wt(z) ∈ Λ^{+}. This will always be the case, and in fact wt(z^{+}) = f.wt(z). There is exactly one such f of minimal length, showing why f and z^{+} are well-defined.

Lemma 3.1.4 Let z ∈ I(n, r) be such that i < j ⇒ z_{i} ≤ z_{j}. In particular, z is of parabolic form; let the associated parabolic subgroup be W_{Z}. Then:

(i) τ_{z,z^{+}} × τ_{z^{+},z} = v^{∗}τ_{z,z}; (ii) There exists u ∈ S_{v}(n, r) satisfying u.ξ_{z^{+},z} = v^{∗}ξ_{z}. Proof It is immediate that if z is of parabolic form, then so is z^{+}, and the stabiliser of z^{+} is precisely W_{Z}. We then know from the remarks following the definition of the τ_{i,j} that τ_{z^{+},z} = v^{∗}ξ_{z^{+},z}, for some integer ∗, and that τ_{z,z} = v^{∗}ξ_{z,z} for some integer ∗. Furthermore, we know that τ_{z,z^{+}} corresponds to some element in S_{v}(n, r). This establishes the proof of (ii) assuming (i), so it remains to prove (i).

To prove (i), it is enough, by the results of Corollary 2.2.5, to prove that the only relaxed basis element occurring in τ_{z,z^{+}} × τ_{z^{+},z} is that corresponding to τ_{z,z}, and that it occurs with a coefficient which is a power of v.

The only relaxed basis element occurring with nonzero coefficient in τ_{z^{+},z} which can contribute to a relaxed basis element in the product is u_{z^{+},z}. This follows from Corollary 2.2.5 and the fact that τ_{z^{+},z} = v^{∗}.ξ_{z^{+},z}. Lemma 2.2.4 tells us that the coefficient of u_{z^{+},z} is a power of v.

The problem now reduces further—it is now enough to show that if u_{a,z^{+}} occurs in τ_{z,z^{+}} with nonzero coefficient then a = z and the coefficient is a power of v.

Using Lemma 3.1.3 (i), the definition of τ, and the action of T_{y} (where y ∈ W_{Z}) on u_{z,z^{+}}, we find that

τ_{z,z^{+}} = v^{∗}.u_{z,z^{+}}. ( ∑_{w∈D_{Z}} D_{w}^{z^{+}}) .

From our study of the properties of D_{w}^{z^{+}}, we see that any basis element occurring with nonzero coefficient in u_{z,z^{+}}D_{w}^{z^{+}}, where w is as in the above sum, is of form u_{b,z^{+}.w}. The implication of this is that the only basis elements of the desired form come from the case w = 1 in the sum. This is enough to complete the proof of the Lemma.

Remark on Lemma 3.1.4 In the classical case, this result is obvious (see [W]).

Proposition 3.1.5 Any codeterminant is a ℤ[q, q^{−1}]-linear combination of codeterminants of dominant shape. Proof We know that u.ξ_{z^{+},z} = ξ_{z} for some u ∈ S_{v}(n, r), because v is a unit. Since

S_{v}(n, r) ≅ S_{q}(n, r) ⊕ v.S_{q}(n, r)

as a ℤ[q, q^{−1}] module, we can take u to be in S_{q}(n, r), by omitting irrelevant terms.

Since u.ξ_{z^{+},z} = u.ξ_{z^{+}}.ξ_{z^{+},z}, we see that u.ξ_{z^{+}} is a ℤ[q, q^{−1}]-linear combination of basis elements of the form ξ_{a,z^{+}}. It now follows that ξ_{z} is a ℤ[q, q^{−1}]-linear combination of codeterminants of shape z^{+}, i.e. of dominant shape.

Now consider an arbitrary codeterminant, Y_{i,j}^{wt(z)}, where z can be taken to be as in the statement of Lemma 3.1.4 (and so in particular, z = ℓ(z)). This is equal to ξ_{i,z}ξ_{z}ξ_{z,j}. Using the first part of the proof, we can now rewrite this as

∑_{a} μ_{a} (ξ_{i,z}ξ_{a,z^{+}}) (ξ_{z^{+},z}ξ_{z,j}) ,

where the coefficients μ_{a} lie in ℤ[q, q^{−1}]. Expanding the parentheses using the product rule, we can finally rewrite the sum as a ℤ[q, q^{−1}]-linear combination of codeterminants of dominant shape, as required.

3.2. The Quantized Straightening Formula

The next main result will be the proof of the quantum analogue of Woodcock’s straightening formula for codeterminants. The aim is to express an arbitrary codeterminant as a linear combination of standard codeterminants. Proposition 3.1.5 shows that it is enough to be able to express an arbitrary codeterminant of dominant shape as a linear combination of standard codeterminants.

Let λ ∈ Λ^{+}(n, r), ℓ = ℓ(λ) and i ∈ I′_{λ}\I_{λ}. Note that ℓ is of parabolic form; denote the associated parabolic subgroup by W_{L}. We shall construct another element j ∈ I(n, r) by analysis of the tableau T = T_{i}^{λ}, in exactly the same way as Woodcock [W]. (The symbols i, j and λ will remain fixed throughout the next few lemmas.)

Denote by [λ] the Young diagram of λ, considered as a subset of ℕ^{2}. By the choice of i, T cannot be standard, and so we know there is some (b, c) ∈ [λ] satisfying T (b, c) ≥ T (b + 1, c). Let (b, c) be lexicographically minimal with respect to this property. Denote by [λ]_{a} the a-th row of [λ], and define X_{a} and Y_{a} so that [λ]_{a} = X_{a} ˙∪ Y_{a} (a ∈ n), where

X_{a} = [λ]_{a}, Y_{a} = ∅, if a < b;
X_{b} = {⟨b, h⟩ : 1 ≤ h < c}  Y_{b} = [λ]_{b}\X_{b};
X_{a} = {x ∈ [λ]_{a} : T (x) ≤ T (y) for all y ∈ Y_{a−1}}, Y_{a} = [λ]_{a}\X_{a}, if a > b.

Put ν_{a} = |X_{a} ˙∪ Y_{a−1}| for a ∈ n + 1, and define X_{n+1} = Y_{0} = ∅. Let a_{1}, …, a_{m} be the elements of {a ∈ n + 1 : ν_{a} ≠ 0}, ordered so that α < β ⇒ ν_{a_{α}} ≥ ν_{a_{β}}.

Note that m ≤ n: this is clear if X_{b} = ∅. If X_{b} ≠ ∅, each entry of T in X_{b+1} is at least b + 1 by the minimality of (b, c). By induction, each of the elements of Y_{a} is at least a + 1, so Y_{n} = ∅.

We are now ready to define j ∈ I by the property

T_{j}^{λ}(x) = α if x ∈ X_{a_{α}} ˙∪ Y_{a_{α}−1}.

Let μ = wt(j). By the choice of ordering on the a_{α}, we have μ ∈ Λ^{+}, and in fact μ ≻ λ since ν_{α} = λ_{α} if α < b, while ν_{b+1} > λ_{b}.

We also define j′ ∈ I by the property that

T_{j′}^{λ}(x) = α if x ∈ X_{α} ˙∪ Y_{α−1}.

Note that we may choose the ordering on a_{1}, …, a_{m} above in such a way that j^{′+} = j.

Example Suppose n = 6, r = 18 and

T_{i}^{λ} = 1 2 3 5 1 2 3 6 1 3 3 1 3 5 2 3 2 2 .

Then

T_{j}^{λ} = 1 3 2 4 1 3 2 5 1 2 2 1 2 4 1 2 1 1

and

T_{j′}^{λ} = 1 2 3 4 1 2 3 5 1 3 3 1 3 4 1 3 1 1 .

The element j is given by

\underbrace{1, 1, 1, 1, 1, 1, 1}_{X_{1}}, \underbrace{3, 3}_{X_{2}}, \underbrace{2, 2, 2}_{Y_{2}}, \underbrace{2, 2, 2}_{X_{3}}, \underbrace{4}_{Y_{3}}, \underbrace{4}_{X_{4}}, \underbrace{5}_{Y_{4}},

where the sections of j corresponding to each nonempty X_{a} and Y_{a} have been marked. Similarly, the element j′ is given by

\underbrace{1, 1, 1, 1, 1, 1, 1}_{X_{1}}, \underbrace{2, 2}_{X_{2}}, \underbrace{3, 3, 3}_{Y_{2}}, \underbrace{3, 3, 3}_{X_{3}}, \underbrace{4}_{Y_{3}}, \underbrace{4}_{X_{4}}, \underbrace{5}_{Y_{4}} .

The weights λ and μ are given by

λ = (7, 5, 4, 2, 0, 0)

and

μ = (7, 6, 2, 2, 1, 0).

The following lemma is very useful for reducing questions about q-codeterminants to questions about their classical analogues. Lemma 3.2.1 Suppose a, b, d, e ∈ I(n, r). Assume that a_{1} ≤ ⋯ ≤ a_{r} and b_{1} ≤ ⋯ ≤ b_{r} (which implies that ⟨a, b⟩ and ⟨b, a⟩ are H-relaxed).

(i) Write

ξ_{a,b}ξ_{d,e} = ∑_{a′∼a} c_{a′,e}ξ_{a′,e}.

Then the elements c_{a′,e} lie in ℕ[q]. (ii) Write

ξ_{e,d}ξ_{b,a} = ∑_{e′∼e} c′_{e′,a}ξ_{e′,a}.

Then the elements c′_{e′,a} lie in ℕ[q].

Proof It is clear from the product rule for the q-Schur algebra given in [BLM, Proposition 1.2] that a product of two basis elements is a ℤ[q]-combination of other basis elements. To prove both parts of the lemma, it therefore suffices to show that the coefficients occurring lie in ℕ[v, v^{−1}].

We know from [DJ3, Theorem 1.11] that there is an anti-automorphism of the q-Schur algebra which sends an element ξ_{α,β} to ξ_{β,α}. This implies that it is enough to prove (i), because (ii) is an immediate consequence of it.

We now concentrate on (i). We may assume that ⟨d, e⟩ is an H-relaxed pair. Define A to be the matrix satisfying e_{A} = ξ_{d,e} and define s_{A} and m_{A} as in §2.2. The parabolic subgroup of W which is the stabiliser of e is denoted by W_{e}, and we write D_{d,e} for the set of distinguished right coset representatives corresponding to the parabolic subgroup which is the stabiliser of the pair (d, e).

Any relaxed basis element u_{α,β} occurring in the product Π_{0} := ξ_{a,b}ξ_{d,e} must have β = e and α ∼ a. Now Corollary 2.2.5 implies that the coefficient of u_{α,β} in Π_{0} is the same as the coefficient in the product ξ_{a,b}x_{d,e}, where x_{d,e} ∈ T^{r}(M_{n}) is given by

x_{d,e} := v^{s_{A}−m_{A}}u_{d,e}. ( ∑_{w∈W_{e}∩D_{d,e}} T_{w}) .

Because we assumed that ⟨d, e⟩ was H-relaxed, we find by induction on left subwords of w and iteration of equation (1) in §3.0 that if T_{w} is one of the elements in the sum, then

u_{d,e}.T_{w} = v^{∗}u_{dw,ew},

for some integer denoted by ∗. It follows that x_{d,e} is an ℕ[v, v^{−1}] combination of basis elements of tensor matrix space.

Now consider e_{B} = ξ_{a,b} as an element of tensor matrix space. We find from Theorem 2.2.7 that

ξ_{a,b} = v^{s_{B}−m_{B}}u_{a,b}. ( ∑_{w∈D_{a,b}} T_{w}) .

We find that for each w in the sum,

u_{a,b}.T_{w} = v^{∗}u_{aw,bw}.

This is proved using the hypotheses on the pair ⟨a, b⟩, induction on left subwords of w, and iteration of equation (1) in §3.0. An argument similar to the one in the preceding paragraph shows that ξ_{a,b} is an ℕ[v, v^{−1}]-combination of basis vectors of tensor matrix space.

It is now immediate from Corollary 2.2.5 that the elements c_{a′,e} lie in ℕ[v, v^{−1}], which completes the proof.

Definitions Let Π_{0} be the element of S_{q}(n, r) given by

ξ_{i,j′}ξ_{j′,l},

and let Π_{1} be the element of S(n, r) given by the same expression.

Let Π_{2} be the element of S(n, r) given by

ξ_{i,j}ξ_{j,l}.

Lemma 3.2.2 Express Π_{0} in terms of the basis elements of the q-Schur algebra {ξ_{α,β}} (where the pairs ⟨α, β⟩ are H-relaxed) by writing

Π_{0} = ∑_{i′∈I′_{λ}} c_{i′,l}ξ_{i′,l},

where c_{i′,l} ∈ ℤ[q]. Then c_{i,l} is a power of q and i′ ⪰ i whenever c_{i′,l} is nonzero.

Proof Note that the product Π_{0} = ξ_{i,j′}ξ_{j′,l} satisfies the hypotheses of Lemma 3.2.1 (ii). Write c′_{i′,l} for the integer which is the specialisation as q is replaced by 1 of c_{i′,l}. It is now enough in the light of Lemma 3.2.1 to prove that the coefficient c′_{i,l} in Π_{1} is 1 and that i′ ⪰ i whenever c′_{i′,l} ≠ 0, because c′_{i,l} is equal to 1 if and only if c_{i,l} is a (nonnegative) power of q, and c′_{i′,l} is nonzero if and only if c_{i′,l} is nonzero. (This is true because the coefficients in the statement of Lemma 3.2.1 all lie in ℕ[q].)

It is immediate from Schur’s product rule and the construction of j and j′ that Π_{1} and Π_{2} are equal. The element Π_{2} was studied in [W], where it was proved that the coefficient of ξ_{i,l} in Π_{2} was equal to 1 and that any basis element ξ_{i′,l} appearing in Π_{2} satisfied i′ ⪰ i.

The proof now follows.

Lemma 3.2.3 The element Π_{0} can be written as an A-linear sum of q-codeterminants of shape μ. Proof We see from Lemma 3.1.4 (i) that

τ_{j′,j}τ_{j,j′} = v^{∗}τ_{j′,j′}

for some integer denoted by ∗. It follows that we can write

Π_{0} = ξ_{i,j′}ξ_{j′,l} = ξ_{i,j′}ξ_{j′,j′}ξ_{j′,l} = v^{∗} (ξ_{i,j′}τ_{j′,j}) (τ_{j,j′}ξ_{j′,l}) .

Now we expand each pair of parentheses in terms of the basis ξ_{α,β} to obtain an A-linear sum of q-codeterminants of shape μ = wt(j).

Lemma 3.2.4 Let λ ∈ Λ^{+}(n, r), ℓ = ℓ(λ), and i ∈ I′_{λ}\I_{λ}. Then there exist elements c_{i′,ℓ} ∈ A and j ∈ I(n, r) such that μ = wt(j) ∈ Λ^{+}, μ ≻ λ, and

ξ_{i,ℓ} = q^{∗}Π_{0} + ∑_{{i′∈I{′ \atop λ} \atop i′≻i}} c_{i′,ℓ}ξ_{i′,ℓ}.

Thus

ξ_{i,ℓ} = ∑_{{a∼i \atop b∼ℓ}} c′_{a,b}Y_{a,b}^{μ} + ∑_{{i′∈I{′ \atop λ} \atop i′≻i}} c_{i′,ℓ}ξ_{i′,ℓ},

where the scalars c and c′ lie in A. Proof The elements j and μ are constructed from i and λ as previously, where it was also shown that μ ∈ Λ^{+} and μ ≻ λ. The first assertion follows from rearranging the expression for Π_{0} given by Lemma 3.2.2, and using the fact that the coefficient of ξ_{i,ℓ} is a power of q (i.e. a unit). The second assertion follows from the first by re-expressing Π_{0} as an A-linear sum of codeterminants of shape μ, as explained in Lemma 3.2.3.

We require the following well-known property of q-Schur algebras in the proof of the main result. Proposition 3.2.5 There is an anti-isomorphism ∗ : S_{q}(n, r) → S_{q}(n, r) given by either

\begin{aligned}(ϕ_{λ,μ}^{d})^{∗} & = ϕ_{μ,λ}^{d^{−1}}, \\ (e_{A})^{∗} & = e_{A^{⊤}}, \\ or (ξ_{a,b})^{∗} & = ξ_{b,a}. \end{aligned}

Thus (Y_{a,b}^{ν})^{∗} = Y_{b,a}^{ν}.

Proof For the proof of this with respect to the ϕ, see [DJ3, Theorem 1.11]. The other two equivalent definitions follow from the equivalence of the three bases.

The last assertion is immediate from the definition of codeterminants.

Definition Following [W], we define

B := {(λ, i, j) : λ ∈ Λ^{+}, i, j ∈ I′_{λ}},

and we partially order B by (λ, i, j) ⪯ (μ, i′, j′) if and only if λ ≺ μ or (λ = μ, i ⪯ i′ and j ⪯ j′).

We can now state and prove the main theorem. Theorem 3.2.6 (the Straightening Formula for Quantized Codeterminants)

Let Y_{i,j}^{λ} be any codeterminant. Then Y_{i,j}^{λ} is a ℤ[q, q^{−1}]-linear combination of standard codeterminants. Proof By Proposition 3.1.5, it is enough to prove the theorem in the case where λ ∈ Λ^{+}. Since

S_{v}(n, r) ≅ S_{q}(n, r) ⊕ v.S_{q}(n, r)

as a ℤ[q, q^{−1}] module, and all codeterminants Y_{a,b}^{μ} lie in S_{q}(n, r), we see that it is enough to prove the result that Y_{i,j}^{λ} is an A-linear combination of standard codeterminants.

Suppose Y_{i,j}^{λ} is not standard. It is enough to show that we can write it as an A-linear combination of codeterminants corresponding to elements of B which are strictly higher in the partial order on B, and then use induction on the poset (B, ⪯).

Assume first that i ∉ I_{ℓ}. Construct the elements j ∈ I(n, r) and μ ∈ Λ^{+} as in Lemma 3.2.4, and set ℓ = ℓ(λ).

By definition, Y_{i,j}^{λ} = ξ_{i,ℓ} × ξ_{ℓ,j}. We now substitute for ξ_{i,ℓ} using the formula in Lemma 3.2.4 to obtain

Y_{i,j}^{λ} = (∑_{{a∼i \atop b∼ℓ}} c′_{a,b}Y_{a,b}^{μ} × ξ_{ℓ,j}) + ( ∑_{{i′∈I{′ \atop λ} \atop i′≻i}} c_{i′,ℓ}ξ_{i′,ℓ}ξ_{ℓ,j}) ,

where the scalars c and c′ all lie in A. Next, we can rewrite the expression in the left-hand set of parentheses using the fact that

Y_{a,b}^{μ}ξ_{ℓ,j} = ξ_{a,ℓ(μ)}(ξ_{ℓ(μ),b}ξ_{ℓ,j}),

and then expanding the product ξ_{ℓ(μ),b}ξ_{ℓ,j} as a sum of basis elements ξ_{ℓ(μ),d} by using the product rule. Finally we can write the product Y_{a,b}^{μ}ξ_{ℓ,j} as an A-linear sum of codeterminants Y_{a,d}^{μ}. This gives

Y_{i,j}^{λ} = (∑_{{a∼i \atop d∼j}} c′′_{a,d}Y_{a,d}^{μ}) + ( ∑_{{i′∈I{′ \atop λ} \atop i′≻i}} c_{i′,ℓ}Y_{i′,j}^{ℓ}) ,

where the scalars c′′ and c all lie in A.

Note that we have now succeeded in expressing Y_{i,j}^{λ} as the sum of codeterminants corresponding to strictly higher elements of B, as required.

The other case, where i ∈ I_{λ} but j ∉ I_{λ} follows by symmetry by using Proposition 3.2.5 followed by the preceding argument, then Proposition 3.2.5 again. This completes the proof.

Corollary 3.2.7 The set of standard codeterminants forms a free ℤ[q, q^{−1}]-basis of S_{q}(n, r). Proof The cardinality of this set is \binom{n^{2}+r−1}{r}, because it is the same as the cardinality of the corresponding set in the classical case. (See [G1, §2] for a proof of the latter.) This is the same as the dimension of S_{q}(n, r). It is also known (see [DJ3, Theorem 1.4]) that the elements e_{A} form a free basis for S_{q}(n, r). It is therefore enough to show that each e_{A} is a ℤ[q, q^{−1}]-linear combination of standard codeterminants. Theorem 3.2.6 shows that it is enough to prove that each basis element ξ_{i,j} is equal to a codeterminant. This is clear, because ξ_{i,j} = ξ_{i,j}ξ_{j}.

We end this chapter by giving a brief summary of the steps needed to straighten an arbitrary codeterminant, Y_{i′,k′}^{λ′}.

(1) Find the element z ∈ I(n, r) satisfying a < b ⇒ z_{a} ≤ z_{b} and wt(z) = λ′. Use the identity τ_{z,z^{+}}τ_{z^{+},z} = v^{∗}τ_{z,z} of Lemma 3.1.4 (i) and the techniques of Proposition 3.1.5 to express the codeterminant as a sum of codeterminants of dominant shape.

(2) Straighten each non-standard codeterminant Y_{i,k}^{λ} arising from the previous procedure. If i ∈ I_{λ} but k ∉ I_{λ}, straighten the codeterminant Y_{k,i}^{λ} instead and use Proposition 3.2.5 to “turn the results around” at the end.

(3) We may now concentrate on straightening codeterminants of form Y_{i,k}^{λ}, where λ is a dominant weight and i ∉ I_{λ}. Construct the elements j and j′ from i and λ, as described at the beginning of §3.2. Using the formula for ξ_{i,ℓ} given in Lemma 3.2.4, and the techniques of the proof of Theorem 3.2.6, rewrite Y_{i,k}^{λ} as a sum of codeterminants corresponding to strictly higher elements of the poset B.

(4) Iterate parts (2) and (3) until all codeterminants appearing in the sum are standard.

4. q-Schur Algebras as Quotients of

Quantized Enveloping Algebras

The main aim of this chapter is to investigate the properties of the surjective algebra homomorphism θ = θ_{r} from the quantized enveloping algebra to the q-Schur algebra. In particular, we would like to be able to find preimages of elements of the q-Schur algebra under this map, and we would like a description of ker θ. These problems are solved at the end of this chapter, and are related to the problem of straightening q-codeterminants which was solved in §3.

4.0 Conventions and review of §3

The algebras U, U^{−}, U^{0} and U^{+} used in this chapter refer to the A-algebras U_{A}(gl_{n}), U_{A}^{−}(gl_{n}), U_{A}^{0}(gl_{n}) and U_{A}^{+}(gl_{n}). These were defined in §1.1.1. As usual, the q-Schur algebra S_{q}(n, r) is defined over ℤ[q, q^{−1}] and the v-Schur algebra S_{v}(n, r) is defined over A.

It is worth re-stating the relationship between tableaux and q-codeterminants in a way compatible with the matrix basis {e_{A} : A ∈ Θ_{r}}.

Consider any ordered pair ⟨T, T′⟩ of standard tableaux of the same shape, each consisting of r boxes and having entries in n. We showed in §1.4.2 how to associate to ⟨T, T′⟩ a certain q-codeterminant, e_{A}e_{A′}. An equivalent way of defining this is as follows.

The entry A_{ij} of A is defined to be the number of occurrences of i in the j-th row of the tableau T, or zero if there is no such entry.

The entry A′_{ij} of A′ is defined to be the number of occurrences of j in the i-th row of the tableau T′, or zero if there is no such entry.

The fact that the tableaux are of the same shape forces the product e_{A}e_{A′} to be nonzero, because it is of form ξ_{i,ℓ}ξ_{ℓ,j}, and by using the product rule for S_{q}(n, r), we find that this is nonzero, and hence e_{A}e_{A′} is a codeterminant. We call such a codeterminant a standard codeterminant. This definition agrees with that in §3.

It should be noted that, because the tableaux T and T′ are standard, the matrix A must be lower triangular, and the matrix A′ must be upper triangular.

Example

Suppose n = 3, r = 6,

T = 1 2 3 2 3 3 ,

and

T′ = 1 2 3 1 2 2 .

Then the matrices A and A′ are given by

A = ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{matrix} )

and

A′ = ( \begin{matrix}2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

The rest of this chapter is divided into two sections. In §4.1, we study the restriction of θ to certain subalgebras of U. In §4.2, we use the properties of quantized codeterminants to complete our description of the relationship between U and S_{q}(n, r).

4.1. The restriction of θ to U^{−}, U^{0} and U^{+}

The next aim is to obtain precise and general descriptions for U^{+} ∩ ker θ and for U^{−} ∩ ker θ, concentrating on the first case, because the second case is essentially similar.

Definition Let X be the matrix

( \begin{matrix}0 & c_{1} & c_{2} & c_{3} & ⋯ & c_{n−1} \\ 0 & 0 & c_{n} & c_{n+1} & ⋯ & c_{2n−3} \\ ⋮ & & & & ... \\ 0 & 0 & 0 & 0 & ⋯ & c_{N} \\ 0 & 0 & 0 & 0 & ⋯ & 0 \end{matrix} ) .

We define

y_{X,r} := ∑_{λ_{1}+···+λ_{n}+|X|=r} [(X, λ_{1}, …, λ_{n})] ,

summed over all sets of nonnegative integers λ_{1}, …, λ_{n}, where the matrix (X, λ_{1}, …, λ_{n}) is given by X + diag(λ_{1}, …, λ_{n}), and |X| denotes the sum of the entries in X.

It is known that, for 1 ≤ i ≤ n − 1, E_{i} maps under θ to y_{X,r}, where X is the matrix E_{i,i+1}, having 1 in (i, i + 1) place and zeros elsewhere.

In order to prove the main result of this section, we will rely on the the following lemma, which is a simple corollary of a lemma by Beilinson, Lusztig and MacPherson. Lemma 4.1.1 Let X be a strictly upper triangular n×n matrix, with |X| ≤ r. Then θ(E_{h})×y_{X,r} is given by

(1 − δ_{|X|,r})v^{β(h+1)}[X_{h,h+1} + 1]y_{X+E_{h,h+1},r} + ∑_{p∈[h+2,n];X_{h+1,p}≥1} v^{β(p)}[X_{h,p} + 1]y_{X+E_{h,p}−E_{h+1,p},r}

where, as usual, E_{i,j} is the matrix with 1 in the (i, j) place, and 0 everywhere else.

Here,

β(p) := ∑_{j>p}(X_{h,j} − X_{h+1,j}).

Let X be a strictly lower triangular n × n matrix. Then θ(F_{h}) × y_{X,r} is given by

(1 − δ_{|X|,r})v^{β′(h)}[X_{h+1,h} + 1]y_{X+E_{h+1,h},r} + ∑_{p∈[1,h−1];X_{h,p}≥1} v^{β′(p)}[X_{h+1,p} + 1]y_{X−E_{h,p}+E_{h+1,p},r}.

Here,

β′(p) := ∑_{j<p}(X_{h+1,j} − X_{h,j}).

Proof This follows from [BLM, Lemma 3.4 (a2)], using the formula for θ(E_{i}).

Definition

Let X be an n × n matrix with entries in ℤ. We say X has the property P (i, j) (where i and j are integers satisfying 1 ≤ i ≤ j < n) if it satisfies one of two conditions. If row j + 1 of X consists entirely of zeros, then X has the property P (i, j). Otherwise, let p be minimal such that X_{j+1,p} is nonzero. If for each integer s such that i < s < j + 1, X_{s,t} = 0 for t ≥ s + p − j, then X has the property P (i, j). If neither condition holds, then X does not have the property P (i, j).

Example Suppose

X = ( \begin{matrix}3 & 5 & 1 & 0 & 3 & 4 \\ 5 & 2 & 3 & 0 & 3 & 1 \\ 0 & 4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 1 & 3 & 0 & 2 & 4 & 4 \end{matrix} ) .

Then X has the properties P (k, k) for all 1 ≤ k < 6, and also P (1, 3), P (2, 3), P (2, 4) and P (3, 4).

Lemma 4.1.2 Let α = α_{i} + α_{i+1} + ⋯ + α_{j} be a positive root.

Assume X is a strictly upper triangular matrix such that if D is a diagonal n by n matrix with nonnegative integer entries, then X +D has the property P (i, s) for all i < s ≤ j. Then θ(E_{α})×y_{X,r} is given by

(1 − δ_{|X|,r})v^{x(j+1)}[X_{i,j+1} + 1]y_{X+E_{i,j+1},r} + ∑_{p∈[j+2,n];X_{j+1,p}≥1} v^{x(p)}[X_{i,p} + 1]y_{X+E_{i,p}−E_{j+1,p},r}.

Here,

x(p) := ∑_{m>p}(X_{i,m} − X_{j+1,m}).

Proof The proof is by induction on n′ = j − i. The case n′ = 0 is done by Lemma 4.1.1. It now suffices to check that, when i < j and β = α − α_{j}, if the hypothesis works for E_{β} and for E_{j} (by induction), then it works for E_{α} := E_{β}E_{j} − v^{−1}E_{j}E_{β}.

Using the inductive hypothesis applied to E_{β} (since we know any matrix X +D has the property P (i, s) for all i < s ≤ j − 1), we can assume that

θ(E_{β}) × y_{X,r} = (1 − δ_{|X|,r})v^{x′(j)}[X_{i,j} + 1]y_{X+E_{i,j},r} + ∑_{p∈[j+1,n];X_{j,p}≥1} v^{x′(p)}[X_{i,p} + 1]y_{X+E_{i,p}−E_{j,p},r},

where

x′(p) := ∑_{m>p}(X_{i,m} − X_{j,m}).

Rephrasing this informally, and concentrating on one particular term, [Y], of y_{X,r}, we find that a typical term in the action of E_{β} on [Y] is to decrease an entry in the (j, p)-place of Y by 1, for some suitable p, to increase the entry in the (i, p) place by 1 (resulting in a new matrix Y′) and to multiply by

v^{y′(p)}[Y′_{i,p}],

where

y′(p) := ∑_{m>p}(Y_{i,m} − Y_{j,m}).

Similarly, considering the action of E_{j} on [Z] we find that we find that the action of a typical term is, for some suitable p′, to decrease an entry in the (j + 1, p′)-place of Y by 1, to increase the entry in the (j, p′) place by 1 (resulting in a new matrix Z′) and to multiply by

v^{z′(p′)}[Z′_{i,p′}],

where

z′(p′) := ∑_{m>p′}(Z′_{j,m} − Z′_{j+1,m}).

The crucial issue is the extent to which these two actions fail to commute if p ≠ p′. Let us suppose that p ≠ p′. We claim that the property P (as above) forces p ≤ p′. Suppose to the contrary that p > p′. This implies that the matrix Y has nonzero entries in the (j + 1, p′) place and in the (j, p) place. However, the matrix Y is assumed to have property P (i, j), and we are also assuming that row j + 1 is not empty. Let p′′ be minimal such that Y_{j+1,p′′} is nonzero. Then we must have p′′ ≤ p′ < p. Putting s = j in the definition of the property P (which is valid since n′ = j − i > 0), we see that Y_{j,t} = 0 for t ≥ p′′, so in particular, Y_{j,p} = 0. This is a contradiction.

Next, consider a typical term, [A], of y_{X,r}. Of course, A is an upper triangular matrix. Acting E_{β}E_{j} on this matrix, on the left, we obtain the new matrix [A′] (with entries (j, p) and (j + 1, p′) decreased by 1, and entries (i, p) and (j, p′) increased by 1) with coefficient

[A_{i,p} + 1][A_{j,p′} + 1]v^{x},

where

x = ∑_{m>p′}(A_{j,m} − A_{j+1,m}) + ∑_{m>p}(A′_{i,m} − A′_{j,m}),

i.e.

∑_{m>p′}(A_{j,m} − A_{j+1,m}) + ∑_{m>p}(A_{i,m} − A_{j,m}) − 1.

Similarly, computing the action of E_{j}E_{β}, we find that matrix [A′] occurs with coefficient

[A_{i,p} + 1][A_{j,p′} + 1]v^{x′},

where

x′ = ∑_{m>p′}(A_{j,m} − A_{j+1,m}) + ∑_{m>p}(A_{i,m} − A_{j,m}),

which is exactly v times the coefficient with the elements acting the other way round. Thus, in the action of E_{β}E_{j} − v^{−1}E_{j}E_{β}, the above terms cancel out.

The only remaining terms correspond to the situation p = p′. Using similar methods to the above, we find that the coefficient of [A′] in the action of E_{β}E_{j} − v^{−1}E_{j}E_{β} on [A] is

[A_{i,p} + 1][A_{j,p} + 1]v^{x′′},

where

x′′ = ∑_{m>p}(A_{i,m} − A_{j+1,m}).

In fact, A_{j,p} is equal to zero, because the matrix [A] has the property P (i, j) and A_{j+1,p} is nonzero. Hence [A_{j,p} + 1] = 1.

Notice that this argument is essentially independent of the entries on the diagonal of A. It therefore applies equally well to the action of E_{α} on y_{X,r}, thus establishing the inductive hypothesis for E_{α}.

This completes the proof.

Recall from §1.1.2 the ordering defined on Φ^{+} associated with the elements E_{α}. Denote the last root in this list by β_{1}, the second from last by β_{2}, etc.

Proposition 4.1.3 The element

E_{β_{N}}^{(c_{N})} ⋯ E_{β_{1}}^{(c_{1})}

maps under θ to y_{X,r}. If |X| > r, then the given basis element maps to zero. Proof The proof will be by induction on n′ = ∑_{k=1}^{N} c_{k}. The case n′ = 1 is done by Lemma 4.1.2. For the general case, we will prove the equivalent statement that

θ(E_{β_{N}}^{c_{N}} ⋯ E_{β_{1}}^{c_{1}}) = (∏_{k=1}^{N}[c_{k}]!) y_{X,r}

Let p be maximal such that c_{p} is nonzero, and let X′ be the matrix obtained by decreasing the entry in the position corresponding to c_{p} in X by 1. We now need to check that E_{β_{p}} acts on y_{X′,r} to give [c_{p}]y_{X,r}, as expected. If n′ > r + 1 then there is nothing to prove, because y_{X′,r} = 0, by the inductive hypothesis. Otherwise, express β_{p} in the form α_{i} + α_{i+1} + ⋯ + α_{j}. The ordering chosen for the positive roots guarantees that if D is a diagonal n by n matrix with nonnegative integer entries, then the matrix X′ +D has no nonzero entries between rows i+1 and j +1 inclusive, except possibly on the diagonal. This means that X′ + D has the properties P (i, s) for i < s ≤ j, so we can apply Lemma 4.1.2. Since X_{j+1,t} = 0 for t > j + 1, we find that only the first term occuring in Lemma 4.1.2 can appear. If n′ = r, then we find that this term too is zero, making our element map under θ to zero, as expected. Further scrutiny of the matrices X′ and X reveals that X_{i,m} = 0 for m > j + 1; again this is by properties of the ordering chosen on the positive roots. This means that the quantity x(j + 1) occurring in Lemma 4.1.2 is equal to 0, and the inductive step immediately follows.

Corollary 4.1.4 U^{+} ∩ ker θ is generated by those basis elements which map to zero under θ, i.e.

U^{+} ∩ ker θ = ⟨{E_{β_{N}}^{(c_{N})} ⋯ E_{β_{1}}^{(c_{1})} : ∑_{i=1}^{N} c_{i} ≥ r + 1}⟩.

Proof The basis elements which do not map to zero all map to elements with different associated matrices X, and hence their images are linearly independent. The corollary now follows.

Corollary 4.1.5 θ(U^{+}) is the subalgebra of S_{q}(n, r) generated by y_{X,r} for all possible X of zero triangular form. Proof Clearly θ(U^{+}) is the subspace generated by the images of the PBW-type basis elements of U^{+}. These elements either map to zero, or to a multiple of one of the above elements y_{X,r}.

Corollary 4.1.6 The dimension of θ(U^{+}) is given by

\binom{\binom{n}{2} + r}{r}.

Proof Firstly, observe that by elementary properties of Pascal’s triangle,

∑_{i=0}^{r} \binom{\binom{n}{2} + i − 1}{i} = \binom{\binom{n}{2} + r}{r}.

Secondly, recall that the number of compositions of r into m pieces is known to be

\binom{m + r − 1}{r}.

To count the dimension of θ(U^{+}), it suffices to enumerate the possible matrices X, because the corresponding y_{X,r} are linearly independent. The possibilities for X correspond to compositions of integers between 0 and r into n(n − 1)/2 pieces. The result now follows.

Remarks on the behaviour of U^{−} The corresponding results for U^{−} are similar in spirit, and correspond to “rotating the basis matrices by a half turn”.

Recall the ordering on Φ^{+} associated with the elements F_{α}. Denote the last root in this list by γ_{1}, the second from last by γ_{2}, etc. We have the following result. Proposition 4.1.7

Let X be the matrix

( \begin{matrix}0 & 0 & 0 & 0 & ⋯ & 0 \\ c_{N} & 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & & & & ... \\ c_{2n−3} & ⋯ & c_{n+1} & c_{n} & 0 & 0 \\ c_{n−1} & ⋯ & c_{3} & c_{2} & c_{1} & 0 \end{matrix} ) .

We define

z_{X,r} := ∑_{λ_{1}+···+λ_{n}+|X|=r} [(X, λ_{1}, …, λ_{n})] ,

summed over all sets of nonnegative integers λ_{1}, …, λ_{n}, where the matrix (X, λ_{1}, …, λ_{n}) is given by X + diag(λ_{1}, …, λ_{n}), and |X| denotes the sum of the entries in X.

Then the element

F_{γ_{N}}^{(c_{N})} ⋯ F_{γ_{1}}^{(c_{1})}

maps under θ to z_{X,r}. If |X| > r, then the given basis element maps to zero.

The intersection of U^{−} with the kernel of θ is given by

U^{−} ∩ ker θ = ⟨{F_{γ_{N}}^{(a_{N})} ⋯ F_{γ_{1}}^{(a_{1})} : ∑_{i=1}^{N} a_{i} ≥ r + 1}⟩.

Proof This is the same as the proof of the corresponding result for U^{+}, with trivial changes.

We now investigate the case of U^{0} ∩ ker θ. To do this, we require the following result. Proposition 4.1.8

A basis for U^{0} is given by the set

K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] ,

where δ_{i} ∈ {0, 1} and t_{i} ∈ ℕ, and where

\left[{K_{i}; c \atop t}\right] := ∏_{s=1}^{t} \frac{K_{i}v^{c−s+1} − K_{i}^{−1}v^{−c+s−1}}{v^{s} − v^{−s}} .

Proof This follows from [L2, Proposition 2.14, Theorem 4.5].

We wish to investigate the images under θ of these basis elements. We know that

θ(K_{i}) = ∑_{D∈𝔻_{r}} v^{d_{i}}[D],

where D = diag(d_{1}, …, d_{n}), 𝔻_{r} is the set of n×n diagonal matrices with nonnegative integer entries summing to r, and as usual

[D] = v^{− dim O_{D}+dim pr_{1}(O_{D})}e_{D},

but since D is diagonal, it is not hard to see that [D] = e_{D}. Since e_{D} corresponds to ϕ_{λλ}^{1} in the notation of Dipper and James [DJ3], we see that the e_{D} form a set of mutually orthogonal idempotents. Armed with this information, it is easy to work out the images of products of the K_{i}, because the multiplication in the v-Schur algebra is componentwise.

Lemma 4.1.9

θ (K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]) = ∑_{D∈𝔻_{r}} v^{δ_{1}d_{1}+···+δ_{n}d_{n}} \left[{d_{1} \atop t_{1}}\right] \left[{d_{2} \atop t_{2}}\right] ⋯ \left[{d_{n} \atop t_{n}}\right] [D].

Proof Because of the preceding discussion, it is enough to verify the Lemma for

θ (\left[{K_{i}; 0 \atop t_{i}}\right]) .

Recalling the definition of

\left[{K_{i}; 0 \atop t_{i}}\right] ,

we find that it is important to find the images under θ of elements of the form

\frac{K_{i}v^{−s+1} − K_{i}^{−1}v^{−s−1}}{v^{s} − v^{−s}} ,

which, since θ is linear, can easily be seen to be

∑_{D∈𝔻_{r}} \frac{v^{d_{i}−s+1} − v^{−d_{i}+s−1}}{v^{s} − v^{−s}} [D],

in other words

∑_{D∈𝔻_{r}} \frac{[d_{i} − s + 1]}{[s]}[D].

It now follows that

θ (\left[{K_{i}; 0 \atop t_{i}}\right]) = ∑_{D∈𝔻_{r}} \left[{d_{i} \atop t_{i}}\right] [D].

The Lemma now follows easily.

Corollary 4.1.10

a) If ∑_{i=1}^{n} t_{i} = r, then

θ (K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]) = v^{δ_{1}t_{1}+···+δ_{n}t_{n}}[T],

where T ∈ 𝔻_{r} and the diagonal entries of T are t_{1}, …, t_{n}.

b) If ∑_{i=1}^{n} t_{i} > r, then

θ (K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]) = 0.

Proof Note that if d_{i} < t_{i} then

\left[{d_{i} \atop t_{i}}\right] = 0.

It now follows, in case a), that there is only one term in the sum given in Lemma 4.1.9, namely the one given. By a similar argument, we find that in case b) there are no surviving terms in the sum of Lemma 4.1.9.

Corollary 4.1.10 provides us with a large part of U^{0} ∩ ker θ. The rest comes from considering basis elements B of U^{0} satisfying ∑_{i=1}^{n} t_{i} ≤ r, and subtracting off other elements corresponding, via Corollary 4.1.10, to the terms appearing in θ(B), yielding: Corollary 4.1.11

θ (K_{1}^{δ_{1}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] − ∑_{D∈𝔻_{r}} v^{δ_{1}d_{1}+···+δ_{n}d_{n}} \left[{d_{1} \atop t_{1}}\right] ⋯ \left[{d_{n} \atop t_{n}}\right] \left[{K_{1}; 0 \atop d_{1}}\right] ⋯ \left[{K_{n}; 0 \atop d_{n}}\right]) = 0

Proof Use Lemma 4.1.9 and Corollary 4.1.10.

Using the basis of U^{0} described earlier, it is now possible to describe U^{0} ∩ ker θ exactly. Proposition 4.1.12 Let κ_{n,r} be the function on the basis of U^{0} given in Proposition 4.1.8 sending

K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]

to

K_{1}^{δ_{1}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] − ∑_{D∈𝔻_{r}} v^{δ_{1}d_{1}+···+δ_{n}d_{n}} \left[{d_{1} \atop t_{1}}\right] ⋯ \left[{d_{n} \atop t_{n}}\right] \left[{K_{1}; 0 \atop d_{1}}\right] ⋯ \left[{K_{n}; 0 \atop d_{n}}\right] .

Then a basis for U^{0} ∩ ker θ is given by {κ_{n,r}(B)} as B runs through basis elements of the form

K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]

except those where all the δ_{i} are equal to zero and ∑_{i=1}^{n} t_{i} = r. Proof By Corollary 4.1.11, θ(κ_{n,r}(B)) = 0 for all basis elements B. By Corollary 4.1.10, if ∑_{i=1}^{n} t_{i} ≠ r, or ∑_{i=1}^{n} t_{i} = r and not all the δ_{i} are zero, then B occurs as a “lowest term” (since d_{i} ≥ t_{i}) of coefficient 1 in κ_{n,r}(B). In this case, κ_{n,r}(B) is nonzero. This proves that the elements given in the statement of the Proposition are linearly independent in U^{0}.

Also by Corollary 4.1.10, if ∑_{i=1}^{n} t_{i} = r, and all the δ_{i} are zero, then κ_{n,r}(B) = 0. In this case, θ(B) is some diagonal basis element [D], and all such basis elements turn up as the image of exactly one such B. This proves that that the elements {κ_{n,r}(B)} as given in the statement of the proposition span U_{0} ∩ ker θ, as required.

Definition Take as a basis B^{0} of U^{0} the elements

K_{1}^{δ_{1}}K_{2}^{δ_{2}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] ,

where t_{i} ∈ ℕ_{0}, and ∑_{i=1}^{n} t_{i} > r or both ∑_{i=1}^{n} t_{i} = r and all the δ_{i} are equal to zero, together with the elements

K_{1}^{δ_{1}} ⋯ K_{n}^{δ_{n}} \left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] − ∑_{D∈𝔻_{r}} v^{δ_{1}d_{1}+···+δ_{n}d_{n}} \left[{d_{1} \atop t_{1}}\right] ⋯ \left[{d_{n} \atop t_{n}}\right] \left[{K_{1}; 0 \atop d_{1}}\right] ⋯ \left[{K_{n}; 0 \atop d_{n}}\right] ,

where t_{i} ∈ ℕ_{0}, and ∑_{i=1}^{n} t_{i} < r or both ∑_{i=1}^{n} t_{i} = r and at least one δ_{i} is nonzero.

This basis will be useful in §4.2.

Note that we now know that B = B^{−}⊗B^{0}⊗B^{+} is a basis for U. This provides a very convenient context in which to study θ.

4.2. Codeterminants, explicit surjectivity and ker (θ)

We now study some of the properties of U^{≥0}, the subalgebra of U generated by U^{0} and U^{+}, and U^{≤0}, the subalgebra of U generated by U^{0} and U^{−}. We concentrate on U^{≥0}. Proposition 4.2.1 Let A be an upper triangular matrix corresponding to a basis element of S_{q}(n, r). Let t_{i} be the sum of the entries in row i of A. Then the element of U^{≥0} given by

\left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] E_{β_{N}}^{(c_{N})} ⋯ E_{β_{1}}^{(c_{1})}

maps under θ to [A], where the c_{i} correspond to the entries above the diagonal in A according to the order on the positive roots associated with the elements E_{α}. Proof This follows from Proposition 4.1.3, Corollary 4.1.10 and the following fact. In the q-Schur algebra as presented by Dipper and James [DJ3, §2], they show that that

ϕ_{ab}^{d}ϕ_{cc}^{1} = δ_{bc}ϕ_{ac}^{d},

and that

ϕ_{aa}^{1}ϕ_{bc}^{d} = δ_{ab}ϕ_{ac}^{d}.

Recalling the discussion before Lemma 4.1.9, and the correspondence between [A]-basis and ϕ_{λμ}- basis, it becomes clear that [D].[A] = [A] if the diagonal entries of [D] are the row sums of [A], and [D][A] = 0 otherwise. Therefore, the effect of the U^{0} part of the element given in the Proposition is to pick out one term of the y_{X,r} expression corresponding to the U^{+}-part of the element in the Proposition. The matrix B corresponding to the term [B] picked out in this way has the same row sums as A, and is the same above the diagonal, so it must actually be A.

These remarks suffice to prove the Proposition.

Corollary 4.2.2 The image of U^{≥0} under θ is precisely the subspace of S_{q}(n, r) spanned by the upper triangular basis elements. Proof All basis elements in the basis B^{0} ⊗ B^{+} of U^{≥0} map under θ to sums of upper triangular basis elements. We know from Proposition 4.2.1 that we can find an element of U^{≥0} mapping to any desired upper triangular basis element. This completes the proof.

Remark The image of U^{≥0} under θ is, in the classical case, precisely the so-called Borel subalgebra of the Schur algebra, often denoted by S^{+}. Similarly, the image of U^{≤0} is the subalgebra denoted by S^{−} in the literature, and θ(U^{≤0}) is the subalgebra spanned by lower triangular basis elements.

The situation for U^{≤0} is extremely similar. For example, Proposition 4.2.1 becomes:

Let A be a lower triangular matrix corresponding to a basis element of S_{q}(n, r). Let t_{i} be the sum of the entries in column i of A. Then the element of U^{≥0} given by

F_{γ_{N}}^{(c_{N})} ⋯ F_{γ_{1}}^{(c_{1})} \left[{K_{1}; 0 \atop t_{1}}\right] \left[{K_{2}; 0 \atop t_{2}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]

maps under θ to [A], where the c_{i} correspond to the entries below the diagonal in A according to the usual order on the positive roots associated with the elements F_{α}.

We are now in a position to describe the relationship between U(gl_{n}) and S_{v}(n, r). It is known (see [L3, §3.2]) that

U ≅ U^{−} ⊗ U^{0} ⊗ U^{+}.

We will work with the basis B = B^{−} ⊗ B^{0} ⊗ B^{+} for the algebra U.

We find from earlier results that all but finitely many elements of B map to zero under θ. We will show that the elements of B which do not map to zero map to elements of S_{v}(n, r) which are closely related to codeterminants.

Definition If Y_{i,j}^{λ} is a typical q-codeterminant given by the factorisation e_{A}e_{A′}, then denote by Ŷ_{i,j}^{λ} the v-codeterminant given by [A][A′]. Recall that the triple (λ, i, j) is related to the product [A][A′] as follows. The entry A_{a,b} of A is defined to be the number of occurrences of a in the b-th row of the tableau of shape λ corresponding to i. The entry A′_{a,b} of A′ is defined to be the number of occurrences of b in the a-th row of the tableau of shape λ corresponding to j.

We now recall the straightening formula for q-codeterminants. Proposition 4.2.3

Let Y_{i,j}^{λ} be any q-codeterminant.

Then Y_{i,j}^{λ} is a ℤ[q, q^{−1}]-linear combination of standard q-codeterminants. The coefficients arising in this expression are unique. Proof This follows from Corollary 3.2.7, where it was shown that the standard q-codeterminants form a free basis for S_{q}(n, r).

We immediately have the following Corollary: Corollary 4.2.4

Let Ŷ_{i,j}^{λ} be any v-codeterminant.

Then Ŷ_{i,j}^{λ} is a ℤ[v, v^{−1}]-linear combination of standard v-codeterminants. The coefficients arising in this expression are unique. Proof This follows because the element Ŷ_{i,j}^{λ} is a power of v times Y_{i,j}^{λ}, and q = v^{2}.

Definitions Define the elements C_{μ,a,b}^{λ,i,j} ∈ A to be such that

Ŷ_{i,j}^{λ} = ∑_{μ,a,b} C_{μ,a,b}^{λ,i,j}Ŷ_{a,b}^{μ},

where the sum is taken over all triples (μ, a, b) such that Y_{a,b}^{μ} is a standard codeterminant.

Let Y_{i,j}^{λ} be a codeterminant, equal to e_{A}e_{A′}. If A is lower triangular and A′ is upper triangular, then we say Y_{i,j}^{λ} and Ŷ_{i,j}^{λ} are distinguished.

Note that any standard codeterminant is distinguished. This follows from the correspondence between the ξ_{i,j} and the e_{A}, and the fact that in a standard tableau, all entries in the i-th row are greater than or equal to i.

Theorem 4.2.5 Any element of the basis B of U which does not map to zero maps to a distinguished v-codeterminant. Every distinguished v-codeterminant is the image of exactly one element of B in a natural way. Proof From our earlier analysis of ker θ ∩ U, we find that the only elements of B which do not map to zero under θ are of form

b = ∏_{α∈Φ^{+}} F_{α}^{(c_{α})} ∏_{i=1}^{n} \left[{K_{i}; 0 \atop t_{i}}\right] ∏_{α∈Φ^{+}} E_{α}^{(b_{α})},

where ∑_{α} c_{α} ≤ r, ∑_{α} b_{α} ≤ r and ∑_{i} t_{i} = r. We know that the element ∏_{i=1}^{n} \left[{K_{i};0 \atop t_{i}}\right] maps under θ to [D], where D = diag(t_{1}, …, t_{n}), and that [D][D] = [D]. Thus θ(b) = θ(b^{−})θ(b^{+}), where b^{−} is given by

∏_{α∈Φ^{+}} F_{α}^{(c_{α})} ∏_{i=1}^{n} \left[{K_{i}; 0 \atop t_{i}}\right] ,

and b^{+} is given by

∏_{i=1}^{n} \left[{K_{i}; 0 \atop t_{i}}\right] ∏_{α∈Φ^{+}} E_{α}^{(b_{α})},

so in particular θ(b^{−}) ≠ 0 ≠ θ(b^{+}). We know from our study of U^{≤0} and U^{≥0} that if θ(b^{−}) ≠ 0 then θ(b^{−}) = [L] for some lower triangular matrix L. Similarly θ(b^{+}) = [U] for some upper triangular matrix U. The number t_{i} is thus the sum of the entries in row i of U, and also the sum of the entries in column i of L. This means that in this case e_{L}e_{U} = ξ_{a,b}ξ_{b,c} for suitable a, b, c ∈ I(n, r), and we see that this quantity is nonzero because Schur’s product rule shows that it is nonzero in the classical case. Therefore θ(b) = [L][U] is a distinguished v-codeterminant.

Conversely, any distinguished v-codeterminant [L][U] such that [L][U] ≠ 0 has a unique element of B which maps to it, by reversing the previous argument. (The integers in the expression of b determine the coefficients of the matrices L and U, and vice versa.)

Definition For each distinguished v-codeterminant Ŷ_{i,j}^{λ} = [L][U], we define the element Υ_{i,j}^{λ} to be that element of B (as above) which maps under θ to [L][U].

Corollary 4.2.6 (kernel of θ) The kernel of θ : U(gl_{n}) → S_{v}(n, r) has as a basis all elements of B which map to zero under θ together with all elements of form

Υ_{i,j}^{λ} − ∑_{μ,a,b} C_{μ,a,b}^{λ,i,j}Υ_{a,b}^{μ},

where Y_{i,j}^{λ} is a distinguished but non-standard codeterminant and Y_{a,b}^{μ} is a standard codeterminant.

Proof This follows from the straightening formula for v-codeterminants applied to the set of distinguished v-codeterminants and the fact that each element of B maps to zero or to a distinguished v-codeterminant.

Corollary 4.2.7 (explicit surjectivity) Let [A] be a basis element of S_{v}(n, r), where e_{A} corresponds to ξ_{i,j}. (We assume, as we may, that j_{a} ≤ j_{b} whenever a < b.) Then the element of U given by

∑_{μ,a,b} C_{μ,a,b}^{wt(j),i,j}Υ_{a,b}^{μ},

where the sum is taken over all (μ, a, b) such that Y_{a,b}^{μ} is standard, maps under θ to [A]. Proof The element [A] is equal to the codeterminant Ŷ_{i,j}^{wt(j)}, because e_{A} = ξ_{i,j} = ξ_{i,j}ξ_{j} and j = ℓ(wt(j)). This is equal, by the straightening formula for v-codeterminants, to

∑_{μ,a,b} C_{μ,a,b}^{wt(j),i,j}Ŷ_{a,b}^{μ}.

The claim now follows from the definition of the elements Υ.

5. q-Weyl modules and q-codeterminants

The finite-dimensional representations of the Schur algebra are largely governed by the behaviour of the so-called Weyl modules. It is well known that each Weyl module has a basis which is indexed by the set of standard tableaux of a fixed shape.

Dipper and James [DJ3] showed that there exist q-analogues of Weyl modules (known as q-Weyl modules) whose top quotients form a complete set of irreducible representations of S_{q}(n, r) over a field. They also described an analogue of the “semistandard basis theorem” (which we shall call the standard basis theorem, since involves the standard tableaux) for Weyl modules.

We show in this chapter that the standard basis theorem for q-Weyl modules can be rederived more simply by using the theory of quantized codeterminants. We shall also make use of the description of the q-Schur algebra as a quotient of the quantized enveloping algebra U(gl_{n}) in §4. This enables us to use the theory of quantum groups to prove the standard basis theorem and provides the link between the representation theory of the quantized enveloping algebra with that of the q-Schur algebra.

As in §4, we work with the integral forms of U(gl_{n}), U^{−}, U^{0} and U^{+}.

We see from Corollary 4.2.6 that U has a basis B of the form B_{0} ˙∪ B_{c}, where B_{0} is a basis of U ∩ker(θ), and the elements B_{c} are bijectively mapped by θ to the elements of S_{v}(n, r) corresponding to standard v-codeterminants. We will use this basis of U in this chapter. (Note that the basis B in §5 differs from the basis B = B^{−} ⊗ B^{0} ⊗ B^{+} that was used in §4.)

5.1 Quantized Left Weyl Modules

We now define a certain family of left U-modules, W_{v}^{λ}, which are parametrised by the elements λ ∈ Λ^{+}. We shall refer to W_{v}^{λ} as the left v-Weyl module of shape λ. It will turn out that W_{v}^{λ} is also a S_{v}(n, r)-module in a natural way, and that our v-Weyl modules are essentially the same as Dipper and James’ q-Weyl modules.

Until further notice, we shall assume that n ≥ r, but this will turn out not to be a serious restriction.

Definitions Let c be an integer satisfying 0 < c ≤ r. Define the element z_{c} of V^{⊗c} to be

∑_{π∈S_{c}}(−v)^{−ℓ(π)}e_{π(1)} ⊗ ⋯ ⊗ e_{π(c)}.

We define the left U-module M_{c} as U.z_{c}.

Lemma 5.1.1 The module M_{c} is a highest weight U-module with highest weight vector z_{c}. That is,

(a) E_{i}.z_{c} = 0 for all i;

(b) K_{i}.z_{c} = v^{m(i)}z_{c} for all i and for some function m (called the highest weight of the module) from {1, …, n} to ℤ;

(c) M_{c} = U^{−}z_{c}. Proof We first note from the nature of the comultiplication Δ that if i ≥ c then E_{i}.e_{h} = 0 for h ≤ c, and hence, by the properties of Δ^{(c−1)}, E_{i}.z_{c} = 0. We may therefore assume that i < c.

We define an equivalence relation, ∼, on the tensors b appearing in the definition of z_{c}, by stipulating that b ∼ b′ if and only if b = b′ or b′ can be obtained from b by exchanging the positions of e_{i} and e_{i+1} as they appear in b. (Notice that they must both appear, since i < c.) Note that each ∼-equivalence class has exactly two elements. It is now enough to show that E_{i} acts as zero on elements of V^{⊗c} of the form

(−v)^{−ℓ(π)}e_{π(1)} ⊗ ⋯ ⊗ e_{π(c)} + (−v)^{−ℓ(σ)}e_{σ(1)} ⊗ ⋯ ⊗ e_{σ(c)},

where the tensors occurring with nonzero coefficients are related via ∼, and ℓ(σ) = ℓ(π) + 1. This follows immediately from the definition of the comultiplication. As an illustration of the principle involved, we find the action of E_{1} on z_{2}.

E_{1}.z_{2} = E_{1}.(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1})
= (1 ⊗ E_{1}).(e_{1} ⊗ e_{2}) − (E_{1} ⊗ K_{1}K_{2}^{−1}).(v^{−1}e_{2} ⊗ e_{1})
= e_{1} ⊗ e_{1} − v.v^{−1}e_{1} ⊗ e_{1}
= 0.

This deals with the proof of (a).

Notice that z_{c} is a sum of multiples of certain basis elements b of V^{⊗c}. The number of times a particular e_{h} appears in such a basis element b does not depend on which b we choose; this is immediate from the definition of z_{c}. By using the comultiplication Δ on K_{i} and acting it on b, we find that K_{i} acts on b via

K_{i}.b = v^{m(i)}b,

where m(h) is the number of times e_{h} occurs in the tensor b. This also proves that K_{i}.z_{c} = v^{m(i)}z_{c}, which is enough to prove (b).

The proof of (c) is immediate from (a) and (b) and the fact that U ≅ U^{−} ⊗ U^{0} ⊗ U^{+}.

Definitions Let λ ∈ Λ^{+}. Define z_{λ} to be the element of V^{⊗r} given by

z_{c_{1}} ⊗ ⋯ ⊗ z_{c_{s}},

where c_{h} = λ′_{h}, and s is maximal subject to the condition that λ′_{s} ≠ 0.

Define W_{v}^{λ}, the left v-Weyl module of shape λ, to be U.z_{λ}.

Lemma 5.1.2 The module W_{v}^{λ} is a highest weight U-module of highest weight λ with highest weight vector z_{λ}. Proof This follows from Lemma 5.1.1. We act the E_{i} on z_{λ} via Δ^{(s−1)}, and Lemma 5.1.1 shows that this is zero.

To see that K_{i}.z_{λ} = v^{λ(i)}z_{λ}, we use the techniques of Lemma 5.1.1 (b) and count the number of e_{h} appearing in each tensor b of z_{λ}, for each h.

The fact that W_{v}^{λ} = U^{−}.z_{λ} is now clear.

Definition Let B_{λ} be the subset of B_{c} consisting of all elements of the form

Υ = F_{γ_{N}}^{(c_{N})} ⋯ F_{γ_{1}}^{(c_{1})}\left[{K_{1}; 0 \atop λ_{1}}\right] ⋯ \left[{K_{n}; 0 \atop λ_{n}}\right].

Here, the λ_{i} are the parts of the (fixed) element λ ∈ Λ^{+}, and the c_{i} are nonnegative integers. As we have assumed that Υ ∈ B_{c}, we have certain restrictions on the values of the c_{i}. Denoting by c_{j,i} the element c_{h} corresponding to the root vector F_{α(i,j)}, it can be shown (from the definition of B_{c}) that necessary and sufficient conditions are that

c_{j,i} + (c_{j+1,i} − c_{j+1,i+1}) + ⋯ + (c_{n,i} − c_{n,i+1}) ≤ λ_{i} − λ_{i+1}

whenever 1 ≤ i < j ≤ n. Note that a similar condition can be found in [CL, (47)].

Note that if Υ ∈ B_{λ} then θ(Υ) = [A][D] where [A][D] is a standard v-codeterminant, and D = diag(λ_{1}, …, λ_{n}). All standard codeterminants of form [A][D], where [D] is as above, appear as the image of a unique element Υ ∈ B_{λ}.

Lemma 5.1.3 All elements of B\B_{λ} annihilate z_{λ} on the left. Proof It is clear that all elements of B_{0} act as zero on tensor space via Δ, and, in particular, annihilate z_{λ}.

Next, consider elements Υ of B_{c} which map under θ to standard codeterminants [A][A′] where A′ is upper triangular but not diagonal. We find from the proof of Theorem 4.2.5 that Υ is naturally expressible in the form u.u^{+}, where u ∈ U and u^{+} is a monomial (including divided powers) in the elements E_{α}. We see by an easy induction on h(α) that E_{α} is naturally expressible as a homogeneous polynomial in the E_{i} of degree h(α). (For example, E_{α(1,2)} = E_{1}E_{2} − v^{−1}E_{2}E_{1}, which is a homogeneous polynomial in E_{1} and E_{2} of degree 2.) Since we know each E_{i} annihilates z_{λ} (from Lemma 5.1.2), we find that each E_{α} also annihilates z_{λ}. Hence Υ annihilates z_{λ}.

It follows from Corollary 4.1.10 and Theorem 2.2.7 that an element Υ of the form

\left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]

will act as the identity on z_{λ} if t_{h} = λ_{h} for all h, and will annihilate z_{λ} otherwise. (Formulated in the notation of Dipper and James, this means that ϕ_{t,t}^{1}.z_{λ} = δ_{t,λ}z_{λ}.)

These observations suffice to complete the proof.

Definition

Let B′_{λ} be the set of elements

Υ′ = F_{γ_{N}}^{(c_{N})} ⋯ F_{γ_{1}}^{(c_{1})}

such that

Υ = F_{γ_{N}}^{(c_{N})} ⋯ F_{γ_{1}}^{(c_{1})}\left[{K_{1}; 0 \atop λ_{1}}\right] ⋯ \left[{K_{n}; 0 \atop λ_{n}}\right] ∈ B_{λ}.

Proposition 5.1.4 The set {b.z_{λ} : b ∈ B_{λ}} is a free A-basis for W_{v}^{λ}. Proof The fact that the set given is a spanning set follows from the definition W_{v}^{λ} = U.z_{λ}, the fact that B is an A-basis for U and Lemma 5.1.3.

To prove independence, we compare this situation with its classical counterpart, which was studied in [CL].

We know (see e.g. the proof of Lemma 5.1.3 above) that the element

\left[{K_{1}; 0 \atop λ_{1}}\right] ⋯ \left[{K_{n}; 0 \atop λ_{n}}\right]

of U acts as the identity on z_{λ}. This implies that the family {b.z_{λ} : b ∈ B′_{λ}} is equal (as a subset of V^{⊗r}) to the family {b.z_{λ} : b ∈ B_{λ}}.

By specialising v to 1, and comparing with [CL, p. 218], we find that the set {b.z_{λ} : b ∈ B′_{λ}} is (canonically) equal to Carter and Lusztig’s free ℤ-basis for the classical left Weyl module. (The order Carter and Lusztig impose on their root vectors in U^{−} happens to agree exactly with ours.)

Now consider the quantized case again. Suppose

( ∑_{i∈B_{λ}} c_{i}Υ_{i}) .z_{λ} = 0,

where the c_{i} lie in A. Suppose the c_{i} are not all 0. We may assume (by multiplying throughout by a suitably large power of v) that the c_{i} are elements of ℤ[v].

If c_{i}(1) = 0 (i.e. if c_{i} specialises to 0) then v − 1 divides c_{i} over the ring ℚ[v]. Since ℤ[v] is a unique factorisation domain, we can invoke Gauss’ Lemma and argue that v − 1 divides c_{i} over the ring A. This implies that we may assume c_{i}(1) ≠ 0 for some i, otherwise we could divide the equation throughout by v − 1 until this was no longer possible. We now replace v by 1. This contradicts Carter and Lusztig’s basis theorem for Weyl modules. We conclude that the set that we give is indeed a free A-basis.

5.2. Quantized Right Weyl Modules

We can also introduce a second type of Weyl module. This is a right U-module which is very closely related to the Weyl modules of §5.1. Since all the proofs are analogous to those in §5.1, we shall only sketch proofs in this section.

Definitions Let the generators of U act on V on the right via

\begin{aligned}e_{i}E_{j} & = δ_{i,j}e_{i+1}, \\ e_{i+1}F_{j} & = δ_{i,j}e_{i}, \\ e_{i}K_{j} & = v^{δ_{i,j}}e_{i}, \\ e_{i}K_{j}^{−1} & = v^{−δ_{i,j}}e_{i}, \end{aligned}

It is easy to check that this definition extends (by multiplication in U and linearity) to give a well-defined action of U on V. This can be checked from the relations in U, or from the fact that e.u = e.θ_{1}(u) where θ_{1} : U → S_{v}(n, 1). Extend this action to a right action of U on V^{⊗r} via the comultiplication Δ.

We now define _{v}^{λ}W, the right v-Weyl module of shape λ, to be z_{λ}.U.

Lemma 5.2.1 The module _{v}^{λ}W is a lowest-weight right U-module with lowest weight λ and lowest weight vector z_{λ}. Proof We need to prove the following:

(a) z_{λ}F_{h} = 0 for all 1 ≤ h < n.

(b) z_{λ}K_{h} = v^{λ(h)}z_{λ}.

(c) _{v}^{λ}W = z_{λ}.U^{+}. The proof of (c) follows from the proofs of (a) and (b), as in the proof of Lemma 5.1.2. The

proof of (b) is almost identical to the corresponding part of Lemma 5.1.2. It remains to prove (a). As in Lemma 5.1.2, we find it is enough to check that F_{h} annihilates z_{c} ∈ V^{⊗c} when acting via Δ. The proof now follows by an analogous argument to that of the proof of Lemma 5.1.1, by considering ∼-equivalence classes of basis elements. We provide a simple example to illustrate this:

z_{2}.F_{1} = (e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}).F_{1}
= (e_{1} ⊗ e_{2}).(K_{1}^{−1}K_{2} ⊗ F_{1}) − (v^{−1}e_{2} ⊗ e_{1}).(F_{1} ⊗ 1)
= v^{−1}e_{1} ⊗ e_{1} − v^{−1}e_{1} ⊗ e_{1}
= 0.

This is enough to complete the argument.

We now wish to find a “standard basis” of the quantized right Weyl modules, _{v}^{λ}W. We denote by W_{λ} and _{λ}W the left (respectively right) classical Weyl modules which are obtained by replacing v by 1. Lemma 5.2.2 As subsets of V^{⊗r}, W_{λ} and _{λ}W are identical. Proof Notice that the action of U^{0} on V in the quantized case on the left is the same as the action on the right. The same holds for the actions of U^{0} on V^{⊗r}. This implies that U^{0} acts on V^{⊗r} on the left in exactly the same way it acts on the right.

The action of E_{h} on V^{⊗r} on the left, in the classical case, is the same as the action of F_{h} on V^{⊗r} on the right. This is true because in the classical case, Δ is cocommutative. Similarly, the action of F_{h} on the left is the same as the action of E_{h} on the right.

In a similar manner, one can prove that the action of E_{α} on the left is the same as the action of F_{α} on the right, and vice versa. It now follows that U.z_{λ} and z_{λ}.U are identical.

Corollary 5.2.3 The module _{λ}W is a free ℤ-module whose rank is equal to the number of standard tableaux of shape λ with entries from 1 to n. Proof This is immediate from Carter and Lusztig’s corresponding classical result for left Weyl modules, and Lemma 5.2.2.

Definition Let _{λ}B be the subset of B_{c} consisting of all elements of the form

Υ = \left[{K_{1}; 0 \atop λ_{1}}\right] ⋯ \left[{K_{n}; 0 \atop λ_{n}}\right]E_{β_{N}}^{(b_{N})} ⋯ E_{β_{1}}^{(b_{1})}.

Here, the λ_{i} are the parts of the (fixed) element λ ∈ Λ^{+}, and the b_{i} are nonnegative integers. As we have assumed that Υ ∈ B_{c}, we have certain restrictions on the values of the b_{i}. Denoting by b_{i,j} the element b_{h} corresponding to the root vector E_{α(i,j)}, it can be shown (from the definition of B_{c}) that necessary and sufficient conditions are that

b_{i,j} + (b_{i,j+1} − b_{i+1,j+1}) + ⋯ + (b_{i,n} − b_{i+1,n}) ≤ λ_{i} − λ_{i+1}

whenever 1 ≤ i < j ≤ n.

Note that if Υ ∈ _{λ}B then θ(Υ) = [D][A] where [D][A] is a standard v-codeterminant, and D = diag(λ_{1}, …, λ_{n}). All standard codeterminants of form [D][A], where [D] is as above, appear as the image of a unique element Υ ∈ _{λ}B.

Note

Let θ_{1} be the homomorphism from U to the matrix ring M_{n} which sends E_{h} to e_{h,h+1}, F_{h} to e_{h+1,h}, K_{h} to v.e_{h,h} and K_{h}^{−1} to v^{−1}.e_{h,h}. Define γ_{r} to be the map from U to T^{r}(M_{n}) given by

γ_{r} := \underbrace{(θ_{1} ⊗ ⋯ ⊗ θ_{1})}_{r times} .

Then we see that our left action of U on V^{⊗r} can be expressed as u.e = γ_{r}(u).e, where the second action is the natural left action of T^{r}(M_{n}) on V^{⊗r}. Similarly, our right action of U on V^{⊗r} can be expressed as e.u = e.γ_{r}(u), where the second action is the natural right action of T^{r}(M_{n}) on V^{⊗r}.

Lemma 5.2.4 All elements of B\_{λ}B annihilate z_{λ} on the right. Proof Let θ_{r} be the usual homomorphism from U to S_{v}(n, r). It is a corollary of the proof of Theorem 2.2.7 that ker γ_{r} = ker(θ), because γ_{r}(u) = γ(θ(u)) for the monomorphism γ described in §2.2. It is clear that the left and right actions of T^{r}(M_{n}) on V^{⊗r} are faithful, so it follows that the kernels of the left and right actions of U on V^{⊗r} are both equal to ker(θ). It now follows that all elements of B_{0} annihilate z_{λ} on the right.

Next, consider elements Υ of B_{c} which map under θ to standard codeterminants [A][A′] where A is lower triangular but not diagonal. We find from the proof of Theorem 4.2.5 that Υ is naturally expressible in the form u^{−}.u, where u ∈ U and u^{−} is a monomial (including divided powers) in the elements F_{α}. We see by an easy induction on h(α) that F_{α} is naturally expressible as a homogeneous polynomial in the F_{i} of degree h(α). Since we know each F_{i} annihilates z_{λ} (from Lemma 5.2.1), we find that each F_{α} also annihilates z_{λ}. Hence Υ annihilates z_{λ}.

It follows from the proof of Lemma 5.1.3 and the similarities between the left and right actions of U^{0} on V^{⊗r} that all elements Υ of the form

\left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right]

will act as the identity on z_{λ} if t_{h} = λ_{h} for all h, and will annihilate z_{λ} otherwise.

These observations suffice to complete the proof.

We are now ready to prove the standard basis theorem for quantized right Weyl modules. Proposition 5.2.5 The set {z_{λ}.b : b ∈ _{λ}B} is a free A-basis for _{v}^{λ}W. Proof The fact that the set given is a spanning set follows from the definition W_{v}^{λ} = z_{λ}.U, the fact that B is an A-basis for U and Lemma 5.2.4.

We now tackle independence. By using an argument based around Gauss’ Lemma (as in Lemma 5.1.4), we find that if the given set is not an A-basis in the quantized case, then the corresponding set in the classical case fails to be a ℤ-basis. However, the set _{λ}B has the same cardinality as the rank of _{λ}W as a ℤ-module (as shown by Corollary 5.2.3). It now follows that the given elements are independent.

Before stating the main result of this chapter, we note that the modules _{v}^{λ}W and W_{v}^{λ} are identical as subsets of tensor space. This is not required for the proof of the standard basis theorem, but we provide a proof for completeness and for interest’s sake.

Definitions

The weight of a basis vector b = e_{i_{1}}⊗⋯⊗e_{i_{r}} of V^{⊗r} is defined to be the n-tuple μ := (μ_{1}, …, μ_{n}) where μ_{h} is the multiplicity with which the vector e_{h} occurs in the basis element. (This implies that K_{h}.b = v^{μ_{h}}.b.)

The subspace V^{μ,r} of V^{⊗r} is that which is spanned by all basis elements of weight μ. We call such a subspace a weight space.

Lemma 5.2.6 If b is an element of V^{μ,r}, then the vectors E_{h}.b, F_{h}.b, b.E_{h} and b.F_{h} all lie in weight spaces.

The vectors F_{h}.b and b.E_{h} lie in the same weight space, and the vectors E_{h}.b and b.F_{h} lie in the same weight space. Proof The first assertion is well-known and is immediate from the definitions of the actions of E_{h} and F_{h} on the left and on the right of V^{⊗r}. For example, if 1 ≤ h < n, b ∈ V^{μ,r} and ν is the weight given by

ν_{i} = \{ \begin{matrix}μ_{i} + 1 & if i = h, \\ μ_{i} − 1 & if i = h + 1, \\ μ_{i} & otherwise, \end{matrix}

then we find that E_{h}.b lies in V^{ν,r}. The other three cases follow a similar format.

The proof of the second assertion also follows quickly.

Definitions The μ-weight space of W_{v}^{λ} (respectively, _{v}^{λ}W) is given by W_{v}^{λ} ∩ V^{μ,r} (respectively, _{v}^{λ}W ∩ V^{μ,r}). If h is an integer satisfying 1 ≤ h < n, then the function f_{h} : Λ(n, r) → ℤ is defined as

f_{h}(μ) := μ_{h} − μ_{h+1}.

We write [e_{i_{1}} ∧ ⋯ ∧ e_{i_{c}}] as shorthand for the quantized exterior power given by

∑_{σ∈S_{c}}(−v)^{−ℓ(σ)}e_{σ(i_{1})} ⊗ ⋯ ⊗ e_{σ(i_{c})},

where the symmetric group is acting by the usual place permutation.

Let T be a tableau with r boxes, and write T (i, j) for the j-th entry in the i-th row of T. Denote the length of the column h of T by C(h), and denote the number of columns of T by c. We define the element L(T) of V^{⊗r} by

L(T) := [e_{T (1,1)} ∧ ⋯ ∧ e_{T (C(1),1)}] ⊗ [e_{T (1,2)} ∧ ⋯ ∧ e_{T (C(2),2)}] ⊗ ⋯ ⊗ [e_{T (1,c)} ∧ ⋯ ∧ e_{T (C(c),c)}]. Define L_{n,r} to be the subspace of V^{⊗r} (over the field ℚ(v)) spanned by all L(T) as T ranges

over all tableaux with r boxes, entries from 1 to n, and strictly increasing columns. If μ ∈ Λ(n, r), we define L_{n,r}^{μ}, the μ-weight space of L_{n,r}, to be L_{n,r} ∩ V^{μ,r}. Notice that any L(T) lies in some weight space of L_{n,r}.

Define the function λ_{a}(T, b) (where a, b ∈ ℕ and T is a tableau) to be the number of occurrences of the number a in columns strictly to the left of column b.

Define the function ρ_{a}(T, b) (where a, b ∈ ℕ and T is a tableau) to be the number of occurrences of the number a in columns strictly to the right of column b.

Define κ_{h} ⊂ ℕ to be the set of numbers c such that column c of T contains an entry h.

Let a, b, c ∈ ℕ and let T be a tableau. The tableau T′ = T (c, a, b) is defined to be that tableau of the same shape as T with the topmost occurrence of the number a in column c replaced by an occurrence of b.

Example Let r = 7, n = 4 and T be the tableau

T = 1 3 4 2 3 3 4 .

Then L(T) is given by

[e_{1} ∧ e_{3} ∧ e_{4}] ⊗ [e_{2} ∧ e_{3}] ⊗ e_{3} ⊗ e_{4},

and [e_{1} ∧ e_{3} ∧ e_{4}] expands to

e_{1} ⊗ e_{3} ⊗ e_{4} − v^{−1}e_{1} ⊗ e_{4} ⊗ e_{3} − v^{−1}e_{3} ⊗ e_{1} ⊗ e_{4} + v^{−2}e_{3} ⊗ e_{4} ⊗ e_{1} + v^{−2}e_{4} ⊗ e_{1} ⊗ e_{3} − v^{−3}e_{4} ⊗ e_{3} ⊗ e_{1}.

The value of λ_{3}(T, 3) is 2, and the value of ρ_{4}(T, 2) is 1.

The tableau T′ = T (2, 3, 4) is given by

T′ = 1 3 4 2 4 3 4 .

We now present multiplication formulae for the left and right actions of the E_{h} and F_{h} on an element L(T).

Lemma 5.2.7 Consider L_{n,r} as a subspace of V^{⊗r} under the usual action of U, and let L(T) ∈ L_{n,r} for some suitable tableau T. Define μ to be the weight such that L(T) is in the μ-weight space of L_{n,r}. Then the following identities hold:

F_{h}.L(T) = ∑_{c∈κ_{h}\κ_{h+1}} v^{−λ_{h}(T,c)+λ_{h+1}(T,c)}L(T (c, h, h + 1)),  (1)
E_{h}.L(T) = ∑_{c∈κ_{h+1}\κ_{h}} v^{ρ_{h}(T,c)−ρ_{h+1}(T,c)}L(T (c, h + 1, h)),  (2)
L(T).F_{h} = ∑_{c∈κ_{h+1}\κ_{h}} v^{−λ_{h}(T,c)+λ_{h+1}(T,c)}L(T (c, h + 1, h)),  (3)
L(T).E_{h} = ∑_{c∈κ_{h}\κ_{h+1}} v^{ρ_{h}(T,c)−ρ_{h+1}(T,c)}L(T (c, h, h + 1)).  (4)

Therefore, the following identities also hold:

v^{f_{h}(μ)−1}F_{h}.L(T) = L(T).E_{h},  (5)
v^{f_{h}(μ)+1}L(T).F_{h} = E_{h}.L(T).  (6)

Proof The proof of (5) follows from (1) and (4) and the fact that for any c ∈ κ_{h}\κ_{h+1} we have

f_{h}(μ) = λ_{h}(T, c) + 1 + ρ_{h}(T, c) − λ_{h+1}(T, c) − ρ_{h+1}(T, c).

The proof of (6) follows from (2) and (3) and the fact that for any c ∈ κ_{h+1}\κ_{h} we have

f_{h}(μ) = λ_{h}(T, c) + ρ_{h}(T, c) − λ_{h+1}(T, c) − 1 − ρ_{h+1}(T, c).

We must now establish the truth of (1), (2), (3) and (4). The actions of E_{h} and F_{h} on L(T) (on the left or on the right) can in some sense be regarded as actions on the columns of T and extended via the comultiplication Δ. The functions λ and ρ are exactly what is needed to make the lemma work in (1), (2), (3) and (4), provided that the lemma remains true for tableaux which consist of a single column, so we concentrate on this case.

We are now assuming that T has one column. Since we assume L(T) is defined, the entries of T strictly increase down the columns. If T contains the number h + 1 but not h, or if it does not contain h + 1 at all, (2) and (3) are easily verified. Similarly if T contains the number h but not h + 1, or if it does not contain h at all, (1) and (4) are easily verified.

The only difficult cases are when T contains both h and h + 1. We wish to show that E_{h} and F_{h} act as zero on the left and on the right. The tableau

T = a ; b ,

where a = h and b = h + 1, illustrates all the important features of the general case. There are four cases to check, which we present below. We assume h = 1 in the examples below for clarity, although this is not important.

E_{1}.(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}) = (1 ⊗ E_{1} + E_{1} ⊗ K_{1}K_{2}^{−1}).(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1})

= e_{1} ⊗ e_{1} − v.v^{−1}e_{1} ⊗ e_{1} = 0

F_{1}.(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}) = (F_{1} ⊗ 1 + K_{1}^{−1}K_{2} ⊗ F_{1}).(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1})

= e_{2} ⊗ e_{2} − v.v^{−1}e_{2} ⊗ e_{2} = 0

(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}).E_{1} = (e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}).(1 ⊗ E_{1} + E_{1} ⊗ K_{1}K_{2}^{−1})

= e_{1} ⊗ e_{1} − v.v^{−1}e_{1} ⊗ e_{1} = 0

(e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}).F_{1} = (e_{1} ⊗ e_{2} − v^{−1}e_{2} ⊗ e_{1}).(F_{1} ⊗ 1 + K_{1}^{−1}K_{2} ⊗ F_{1})

= e_{2} ⊗ e_{2} − v.v^{−1}e_{2} ⊗ e_{2} = 0

These observations suffice to complete the proof.

Proposition 5.2.8 As sets, W_{v}^{λ} = _{v}^{λ}W. Proof Denoting the basic λ-tableau by T, we have z_{λ} = L(T). It is clear from Lemma 5.1.1 (c), Lemma 5.2.1 (c) and the definitions of left and right v-Weyl modules that W_{v}^{λ} = U^{−}.L(T) and _{v}^{λ}W = L(T).U^{+}.

Since U^{−} has a spanning set consisting of monomials (with divided powers) in the F_{i}, viz. F_{i_{s}}^{(c_{s})} ⋯ F_{i_{1}}^{(c_{1})}, we find that a spanning set for W_{v}^{λ} is given by

F_{i_{s}}^{(c_{s})} ⋯ F_{i_{1}}^{(c_{1})}.L(T)

Similarly, _{v}^{λ}W has a spanning set consisting of elements given by

L(T).E_{i_{1}}^{(c_{1})} ⋯ E_{i_{s}}^{(c_{s})}.

To prove equality of the two modules, it is enough to show that

F_{i_{s}}^{(c_{s})} ⋯ F_{i_{1}}^{(c_{1})}.L(T) = v^{∗}L(T).E_{i_{1}}^{(c_{1})} ⋯ E_{i_{s}}^{(c_{s})}

for some suitable integer represented by ∗. Since the divided powers are the same on both sides of the equation, this reduces to proving that

F_{i_{s}}^{c_{s}} ⋯ F_{i_{1}}^{c_{1}}.L(T) = v^{∗}L(T).E_{i_{1}}^{c_{1}} ⋯ E_{i_{s}}^{c_{s}}.

Since L(T) lies in a weight space of L_{n,r} (and hence of each v-Weyl module), the proof of the proposition now follows from repeated application of Lemma 5.2.6 and Lemma 5.2.7, equation (5).

5.3. q-Codeterminants and q-Weyl modules

We are now ready to explain the relationship between q-codeterminants and q-Weyl modules. Definitions

For each λ ∈ Λ^{+}, define the left q-Weyl module of shape λ, W_{q}^{λ}, to be S_{q}(n, r).z_{λ}. For each λ ∈ Λ^{+}, define the right q-Weyl module of shape λ, _{q}^{λ}W to be z_{λ}.S_{q}(n, r). Note that the left and right v-Weyl modules can be recovered by appropriate tensoring over A. Let C^{λ} be the set of standard q-codeterminants e_{A}e_{D} such that D = diag(λ_{1}, …, λ_{n}). Let ^{λ}C be the set of standard q-codeterminants e_{D}e_{A′} such that D = diag(λ_{1}, …, λ_{n}).

Theorem 5.3.1 (the Standard Basis Theorem for q-Weyl Modules)

(a) The set {C^{λ}.z_{λ} : λ ∈ Λ^{+}(n, r)} is a free ℤ[q, q^{−1}]-basis for W_{q}^{λ}.

(b) The set {z_{λ}.^{λ}C : λ ∈ Λ^{+}(n, r)} is a free ℤ[q, q^{−1}]-basis for _{q}^{λ}W. Proof Part (a) is a restatement of Proposition 5.1.4, and part (b) is a restatement of Proposition 5.2.5. To see this, we use Theorem 4.2.5 to explain the relationship between PBW-type elements in U with standard codeterminants in S_{q}(n, r), and also the fact that

S_{v}(n, r) ≅ S_{q}(n, r) ⊕ v.S_{q}(n, r)

as a ℤ[q, q^{−1}] module.

We need to check that our quantized left Weyl modules agree with those of Dipper and James in [DJ3]. When λ ∈ Λ^{+} and n > r, Dipper and James define the (left) q-Weyl module to be the left ideal of S_{q}(n, r) generated by an element which they call z_{λ}. This is not obviously the same as our z_{λ}, although it will turn out that the two are multiples of each other. The Dipper–James definition of z_{λ} is

ϕ_{λω}^{1}T_{w_{λ}}y_{λ′},

where

y_{λ′} := ∑_{π∈S_{λ′}}(−q)^{−ℓ(π)}T_{π}.

The element T_{π} represents the element ϕ_{ωω}^{π} or ξ_{ω.π,ω} of S_{q}(n, r), and can be considered to be a basis element of the Hecke algebra. Thus, in the combinatorial set-up of the q-Schur algebra, the element z_{λ} of Dipper and James is denoted by

ξ_{ℓ(λ),ω}ξ_{ω.w_{λ},w} ∑_{π∈S_{λ′}}(−q)^{−ℓ(π)}ξ_{ω.π,ω}.

The next few results establish the required equivalence.

Lemma 5.3.2 Let i ∈ I(n, r) (where n > r as usual) and suppose i_{p} < i_{p+1} for some p. Let s be the simple transposition (p, p + 1). Then the identity

ξ_{i,ω}ξ_{ω.s,ω} = ξ_{i.s,ω}

holds in S_{q}(n, r). Proof

Let e_{A}, e_{P} and e_{A′} be the basis elements expressed above as ξ_{i,ω}, ξ_{ω.s,ω} and ξ_{i.s,ω} respectively. We first consider the product [A][P] by using the techniques introduced in §2.2. Using the notation in that chapter, we find that the only relaxed basis element of T^{r}(M_{n}) occurring in the product with nonzero coefficient is that corresponding to the basis element [A′]. It arises from the product

v^{−m_{A}}t_{A}.T_{s} × v^{−m_{P}} t_{P} .

Note that the condition i_{p} < i_{p+1} ensures that s ∈ D_{A}.

The nature of ω ensures that the integers m_{A}, m_{P} and m_{A′} are all equal to zero. The hypothesis on i ensures that the p-th and (p + 1)-th positions in the tensor t_{A} are of forms e_{a,p} and e_{b,p+1} respectively, where a < b. Since we have

e_{a,p} ⊗ e_{b,p+1}.T_{s} = e_{b,p+1} ⊗ e_{a,p}

by definition of the action of H on T^{r}(M_{n}), we see that t_{A′} occurs with coefficient 1 in the above product.

We now shift our attention to the product e_{A}e_{P}. We know that e_{B} = v^{s_{B}}[B], where

s_{B} := ∑_{1≤i,j,k,l≤n} B_{i,j}B_{k,l},

where the sum is taken with the restrictions that restrictions that i ≥ k and j < l. From this, we see that s_{P} = 1 and s_{A′} = s_{A} + 1. This establishes the desired result that e_{A}.e_{P} = e_{A′}.

Remarks on Lemma 5.3.2 Notice that the corresponding classical identity is immediate from Schur’s product rule. Also notice that the identity in the Hecke algebra T_{ws} = T_{w}T_{s} (where ℓ(ws) > ℓ(w)) can be considered to be a special case of this result.

We now describe a submodule of S_{v}(n, r) which is isomorphic as a left U-module to T^{r}(V). Define the linear map τ : T^{r}(V) → S_{v}(n, r) by

τ(e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}) := [A_{i}],

where the (a, b)-entry of A_{i} is equal to 1 if a = i_{b} and 0 otherwise. Lemma 5.3.3 The map τ is a canonical monomorphism of left U-modules. Proof Note that as usual, S_{v}(n, r) is made into a U-module via

u.s := θ(u) × s.

Injectivity is clear since the elements [A] form a basis of S_{v}(n, r). The fact that τ is a module homomorphism follows from the multiplication formulae in [BLM, Lemma 3.4]. (In fact, this is the reason we use the comultiplication Δ and not one of the other possible comultiplications.)

We now exhibit the link between tensor space and the q-tensor space of Dipper and James. In [DJ3], q-tensor space is defined to be spanned by elements of the form ϕ_{μ,ω}^{d}. In the combinatorial set-up of S_{q}(n, r), this space is spanned by elements of the form ξ_{i,ω}. Lemma 5.3.4 Let i ∈ I(n, r). Write e_{i} for the tensor e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}. Then the map τ satisfies

τ(e_{i}) = v^{−s(i)}ξ_{i,ω},

where s(i) := |{a < b : i_{a} ≥ i_{b}}|. Proof It is easily checked that e_{A_{i}} = ξ_{i,ω} under the usual identification. Since e_{A} = v^{s_{A}}[A], it is enough, by Lemma 5.3.3, to check that s_{A_{i}} = s(i), but this is immediate.

Lemma 5.3.5 We have

ϕ_{λω}^{1}T_{w_{λ}}y_{λ′} = ∑_{π∈S_{λ′}}(−q)^{−ℓ(π)}ξ_{ℓ(λ).w_{λ}.π,ω}.

Proof It is shown in [DJ3, §3] that w_{λ} is a double S_{λ}–S_{λ′} coset representative with the property that

w_{λ}^{−1}S_{λ}w_{λ} ∩ S_{λ′} = 1.

This implies that if a ∈ S_{λ} and b ∈ S_{λ′} then ℓ(aw_{λ}b) = ℓ(a) + ℓ(w_{λ}) + ℓ(b). Choose π ∈ S_{λ′}. Then ℓ(w_{λ}π) = ℓ(w_{λ}) + ℓ(π) and w_{λ}.π is a distinguished right S_{λ}-coset representative.

We now express w_{λ}.π as a product s_{i_{1}} ⋯ s_{i_{m}} of simple reflections, and use induction on p where 1 ≤ p ≤ m to show that

ξ_{ℓ(λ),ω}.ξ_{ω.w_{λ},w}.ξ_{ω.π,ω} = ξ_{ℓ(λ).w_{λ}.π,ω}.

(The element ϕ_{λ,w}^{1} is equal to ξ_{ℓ(λ),ω} under the identification of the two bases.)

Note that since w_{λ}.π is a distinguished right S_{λ}-coset representative, it is immediate that s_{i_{1}} ⋯ s_{i_{p}} is also a distinguished right S_{λ}-coset representative, where 1 ≤ p ≤ m.

The case p = 1 of the induction is to prove that

ξ_{ℓ(λ),ω} × ξ_{ω.s_{i_{1}} ,ω} = ξ_{ℓ(λ).s_{i_{1}} ,ω}.

This is proved by application of Lemma 5.3.2, which is applicable because s_{i_{1}} is a distinguished right S_{λ}-coset representative, so if we express s_{i_{1}} as a transposition (i_{1}, i_{1} + 1), we find that ℓ(λ)_{i_{1}} < ℓ(λ)_{i_{1}+1}.

The inductive hypothesis is that

ξ_{ℓ(λ).s_{i_{1}} ···s_{i_{p−1}} ,ω} × ξ_{ω.s_{ip} ,ω} = ξ_{ℓ(λ).s_{i_{1}} ···s_{ip} ,ω}.

Since s_{i_{1}} ⋯ s_{i_{p}} is a distinguished right S_{λ}-coset representative, we have

(ℓ(λ).s_{i_{1}} ⋯ s_{i_{p−1}})_{i_{p}} < (ℓ(λ).s_{i_{1}} ⋯ s_{i_{p−1}})_{i_{p}+1}.

Lemma 5.3.2 is now applicable, and this completes the induction.

The main proof now follows from the discussion preceding Lemma 5.3.2.

Proposition 5.3.6 The module W_{q}^{λ} is canonically isomorphic to Dipper and James’s module W_{λ}. Proof It is enough to show that, using the map τ of Lemma 5.3.4 and the identity in Lemma 5.3.5 to make the relevant identifications, our element z_{λ} is equal to a power of v times the element z_{λ} ∈ S_{q}(n, r) of Dipper and James.

Define k to be the number of pairs a < b such that ℓ(λ)_{a} ≥ ℓ(λ)_{b}. Notice that if i is the r-tuple ℓ(λ).w_{λ}.π then the number of pairs a < b such that i_{a} ≥ i_{b} is equal to ℓ(w_{λ}) + ℓ(π) + k, by standard properties of symmetric group actions. We now see, using the function s in Lemma 5.3.4, that

s(ℓ(λ).w_{λ}.π) = ℓ(w_{λ}) + ℓ(π) + k.

We now see from Lemmas 5.3.4 and 5.3.5 that W_{λ} is isomorphic as a left U-module to the submodule of tensor space generated by the vector

∑_{π∈S_{λ′}}(−q)^{−ℓ(π)}v^{k+ℓ(w_{λ})+ℓ(π)}e_{ℓ(λ).w_{λ}.π},

Using the fact that q = v^{2}, we find that this is equal to

v^{k′}. ∑_{π∈S_{λ′}}(−v)^{−ℓ(π)}e_{ℓ(λ).w_{λ}.π},

where k′ = ℓ(ω_{λ}) + k.

This now shows that Dipper and James’s z_{λ} can be canonically identified, by multiplying by v^{−k′}, with our z_{λ}. This means that the modules W_{q}^{λ} and W_{λ} are isomorphic as U-modules, and hence as S_{q}(n, r)-modules, as required.

5.4. Remarks on the case n < r

In the case n ≥ r, it is shown in [DJ3, Theorem 8.8] that each of the q-Weyl modules W_{λ} has a unique maximal submodule, which we denote by M_{λ}. The set of all F_{λ} := W_{λ}/M_{λ} as λ varies over Λ^{+} is shown to be a complete and irredundantly described set of absolutely irreducible representations for S_{q}(n, r) over a field.

Dipper and James also remark that if n < r, the complete set of irreducibles is given by

{e.F_{λ} : λ ∈ Λ^{+}(n, r)},

where now Λ^{+}(n, r) is the set of partitions of r into not more than n pieces, F_{λ} is the top quotient of a q-Weyl module S_{q}(N, r) for some N ≥ r, and e is the sum of all idempotent basis elements e_{D} of S_{q}(N, r) where D is a diagonal matrix satisfying D_{i,i} = 0 for i > n.

This construction relies on some canonical embeddings of the q-Schur algebra. The algebra S_{q}(n, r) embeds in S_{q}(N, r) via the map e_{A} ↦ e_{ϵ(A)}, where the (i, j)-entry of ϵ(A) is given by A_{i,j} if i ≤ n and j ≤ n, and is zero otherwise. It is trivial to check that this is a monomorphism of algebras. Another way of looking at this is to notice that S_{q}(n, r) ≅ eS_{q}(N, r)e. This means that the q-Weyl modules for S_{q}(n, r) come from certain q-Weyl modules for S_{q}(N, r) by restriction.

In a somewhat similar way, the quantized enveloping algebra U(gl_{n}) embeds in U(gl_{N}) by the obvious identification of the generators. From the viewpoint of quantized enveloping algebras, the left quantized Weyl modules W_{v}^{λ} for U(gl_{n}) are restrictions of certain left quantized Weyl modules W_{v}^{λ} for U(gl_{N}). The highest weight vectors are the same in both cases. (We have implicitly embedded λ ∈ Λ^{+}(n, r) in Λ^{+}(N, r) by sending λ to (λ_{1}, …, λ_{n}, 0, …, 0).)

Similar remarks hold for right q-Weyl modules. In this case we find that the complete set of irreducibles for S_{q}(n, r) is

{_{λ}F.e : λ ∈ Λ^{+}(n, r)},

and that the modules _{v}^{λ}W for U(gl_{n}) arise from the corresponding ones for U(gl_{N}) by restriction. In the latter case, the lowest weight vectors are identical.

The various versions of the standard basis theorem now carry over essentially unchanged.

6. Cellular inverse limits of q-Schur algebras

It is known that the classical codeterminants of [G2] cast light on the structure of the Schur algebra as a quasi-hereditary algebra (in the sense of Cline, Parshall and Scott).

In this chapter, we make a more detailed study of quantized codeterminants. In §6.1, we strengthen the straightening result given in §3 by proving that any quantized codeterminant of dominant shape λ can be expressed as a linear combination of quantized codeterminants whose shapes dominate λ, in a sense which we will explain. (An analogous result is known (see [M, Proposition 2.6.9]) in the classical case.) This allows one to obtain results concerning the structure constants of the q-Schur algebra with respect to the basis of standard q-codeterminants.

We also show in §6.2 that quantized codeterminants can be used to illustrate the quasi-hereditary structure of the q-Schur algebra and the structure of the q-Schur algebra as a cellular algebra in the sense of Graham and Lehrer [GL]. We study the properties of ideals spanned by codeterminants with shapes lying in a given ascending saturation, which is an ideal of the poset of weights of the q-Schur algebra under the dominance order. In §6.3, we find that, by quotienting out certain of these ideals, one can show that the v-Schur algebra S_{v}(n, r + kn) has as a quotient an algebra canonically isomorphic to S_{v}(n, r) for any natural number k; we construct the epimorphism explicitly.

In §6.4, we construct, using the epimorphisms in §6.3, an inverse system of v-Schur algebras S_{v}(n, r) with fixed values of n. This turns out to be related in an interesting way to the quantized enveloping algebra U(sl_{n}) and to Lusztig’s algebra U̇ of type A.

As in §4 and §5, we use the integral forms of quantized enveloping algebras, except where otherwise stated.

6.1. The strong straightening result

In §3, it was proved that the set of standard v-codeterminants (respectively q-codeterminants) formed a free ℤ[v, v^{−1}] basis (respectively ℤ[q, q^{−1}] basis) of S_{v}(n, r) (respectively S_{q}(n, r)). We shall prove in this section that the standard codeterminants occurring with nonzero coefficients in the expression for a codeterminant of dominant shape λ have shapes equal to or dominating λ. (This result is well-known in the classical case in the context of bideterminants—see for example [M, Proposition 2.6.9].) It also turns out, although we shall not prove this, that this result remains true for codeterminants of arbitrary shape and a suitable natural extension of the order ▷ to all of Λ(n, r). We shall refer to this result as the strong straightening result.

In the proof of this result, we shall require the following basic fact about tableaux. Lemma 6.1.1 Let T be a standard tableau of shape λ and let b ∈ I(n, r) be such that T = T_{b}^{λ}. Let μ = wt(b). Then λ ⊵ μ. Proof Since T is standard, all the occurrences of i (where 1 ≤ i ≤ n) must occur in the first i rows. The result is now immediate from the definition of the dominance order.

We now obtain a partial result for the strong straightening result. Lemma 6.1.2

a) Let Y_{i,ℓ(λ)}^{λ} be a q-codeterminant of dominant shape (not necessarily standard). Then any standard q-codeterminant Y_{a,b}^{ν} appearing in the expansion of Y_{i,ℓ(λ)}^{λ} in terms of standard qcodeterminants satisfies a ∼ i, b ∼ ℓ(λ) and ν ⊵ λ. If ν = λ then b = ℓ(λ).

b) Let Y_{ℓ(λ),j}^{λ} be a q-codeterminant of dominant shape (not necessarily standard). Then any standard q-codeterminant Y_{a,b}^{ν} appearing in the expansion of Y_{ℓ(λ),j}^{λ} in terms of standard qcodeterminants satisfies a ∼ ℓ(λ), b ∼ j and ν ⊵ λ. If ν = λ then a = ℓ(λ).

Proof We only prove a), because the proof of b) follows by symmetry (i.e. the anti-automorphism of S_{q}(n, r) taking Y_{c,d}^{μ} to Y_{d,c}^{μ}).

The assertions a ∼ i and b ∼ ℓ(λ) are immediate from weight space considerations. Consider a typical standard q-codeterminant occurring in this expression and denote it by Y_{a,b}^{ν}. It follows from Lemma 6.1.1 that ν ⊵ λ, because b is of weight λ and Y_{a,b}^{ν} is standard.

The proof of the last part follows from the observation that ℓ(ν)_{i} ≤ b_{i} for all i, because Y_{a,b}^{ν} is standard. If we assume ν = λ then ℓ(ν) ∼ b, which now forces b_{i} = ℓ(ν)_{i} for all i, as required.

The above lemma does the basic work for proving the strong straightening result, which we now prove in full. Proposition 6.1.3 (the Strong Straightening Result)

Let Y_{i,j}^{λ} be a q-codeterminant (not necessarily standard) with the property that λ ∈ Λ^{+}. All standard q-codeterminants appearing in the expansion of Y_{i,j}^{λ} in terms of the q-codeterminant basis are of form Y_{a,b}^{ν}, where a ∼ i, b ∼ j and ν ⊵ λ. Proof It is clear from weight space considerations that a ∼ i and b ∼ j.

To prove the rest of the proposition, we proceed by induction on the partial order ⊵ on the set Λ^{+}(n, r). The base case concerns the weight λ = (r, 0, …, 0), which can easily be checked to have the property that λ ⊵ μ for any μ ∈ Λ^{+}. This case is trivial because any q-codeterminant of shape λ is automatically standard—this follows from the definitions of standard codeterminants.

We now assume the theorem is true for all q-codeterminants of shape ν, where ν ▷λ. We rewrite Y_{i,j}^{λ} as Y_{i,ℓ(λ)}.Y_{ℓ(λ),j} and apply both parts a) and b) of Lemma 6.1.2 to expand each half of the product in terms of standard q-codeterminants. We now have an expression for Y_{i,j}^{λ} in terms of linear combinations of certain products of pairs of standard q-codeterminants. These are of four types, as follows.

1. Products of a term Y_{i′,ℓ(λ)}^{λ} and a term Y_{a,b}^{ν}, where ν ▷ λ.

2. Products of a term Y_{a,b}^{ν} and a term Y_{ℓ(λ),j′}^{λ}, where ν ▷ λ.

3. Products of a term Y_{i′,ℓ(λ)}^{λ} and a term Y_{ℓ(λ),j′}^{λ}.

4. Products of a term Y_{a,b}^{ν} and a term Y_{c,d}^{ν′}, where ν ▷ λ and ν′ ▷ λ.

Products of type 1 and 2 can be naturally re-expressed as linear combinations of quantized codeterminants of shape ν, where ν ▷ λ as described. For example, in type 1 we re-express

Y_{i′,ℓ(λ)}^{λ}.Y_{a,b}^{ν} = (ξ_{i′,ℓ(λ)}.ξ_{a,ℓ(ν)}).ξ_{ℓ(ν),b},

and expand the parentheses using the product rule for q-Schur algebras. Using the inductive hypothesis, we see that these terms are expressible as sums of standard q-codeterminants of shape μ ⊵ ν ▷ λ.

The case of products of type 3 is very simple, because

Y_{i′,ℓ(λ)}^{λ}.Y_{ℓ(λ),j′}^{λ} = Y_{i′,j′}^{λ},

which is a standard q-codeterminant.

Products of type 4 can be dealt with in the same way as products of type 1 or 2.

This completes the induction and the proof of the proposition.

Remarks

1. The straightening formula given in Theorem 3.2.6 proves that any q-codeterminant of dominant shape λ can be expressed as a sum of standard q-codeterminants of shape μ, where μ ⪰ λ for a certain total order ≻ refining ▷. The strong straightening result is not immediately clear from using that technique.

2. A similar result can be formulated for q-codeterminants of arbitrary shape, but we shall not need this for our purposes.

Using the strong straightening result and the properties of increasing saturations, we are now able to define a class of ideals of the q-Schur algebra, as follows.

Proposition 6.1.4 Let π be an increasing saturation. Then the ideal Y (π) of S_{q}(n, r) spanned by all codeterminants of shape λ ∈ π has as a basis the set of all standard q-codeterminants of shapes μ where μ ∈ π. Proof It is clear that the space spanned by all the q-codeterminants of shape λ, where λ ∈ π, forms a two-sided ideal of S_{q}(n, r), by using the techniques needed to deal with the proofs of cases 1 and 2 in the proof of Proposition 6.1.3. It is immediate from the basis theorem for qcodeterminants that the standard q-codeterminants with shapes lying in π are independent over ℤ[q, q^{−1}]. We now need to show that any q-codeterminant of shape λ ∈ π can be expressed as a ℤ[q, q^{−1}]-linear combination of standard q-codeterminants with shapes lying in π, but this is easily seen from Proposition 6.1.3 and the definition of an increasing saturation.

6.2. Cellular algebras and quasi-hereditary algebras

It turns out that q-codeterminants are well-suited to describing the structure of the q-Schur algebra as a cellular algebra, in the sense of Graham and Lehrer. Using the Schur functor, we find that we can also describe the cellular structure of the Hecke algebras of type A.

We recall the definition of a cellular algebra from [GL]. Definition (Graham, Lehrer) Let R be a commutative ring with identity. A cellular algebra over R is an associative unital algebra, A, together with a cell datum (Λ, M, C, ∗) where

1. Λ is a poset. For each λ ∈ Λ, M(λ) is a finite set (the set of tableaux of type λ) such that

C : ∐_{λ∈Λ} (M(λ) × M(λ)) → A

is injective with image an R-basis of A.

2. If λ ∈ Λ and S, T ∈ M(λ), we write C(S, T) = C_{S,T}^{λ} ∈ A. Then ∗ is an R-linear involutary anti-automorphism of A such that (C_{S,T}^{λ})^{∗} = C_{T,S}^{λ}.

3. If λ ∈ Λ and S, T ∈ M_{λ} then for all a ∈ A we have

a.C_{S,T}^{λ} ≡ ∑_{S′∈M(λ)} r_{a}(S′, S)C_{S′,T}^{λ} (mod A(< λ)),

where r_{a}(S′, S) ∈ R is independent of T and A(< λ) is the R-submodule of A generated by the set

{C_{S′′,T′′}^{μ} : μ < λ, S′′ ∈ M(μ), T′′ ∈ M(μ)}.

Proposition 6.2.1 Let R = ℤ[q, q^{−1}]. Let Λ = Λ^{+}(n, r), ordered by reverse dominance so that λ ≤ μ means λ ⊵ μ. Let M(λ) be the set of standard tableaux of shape λ. Let C be the map taking the pair of elements ⟨S, T ⟩ (where S, T ∈ M(λ)) to the associated standard q-codeterminant. Let ∗ be the anti-automorphism of S_{q}(n, r) sending e_{A} to e_{A^{⊤}}. Then (Λ, M, C, ∗) is a cell datum for A = S_{q}(n, r) over R. Proof Axiom (1) follows from the fact (see Corollary 3.2.7) that the standard q-codeterminants give a free ℤ[q, q^{−1}]-basis for A. It is easy to check from the definition of q-codeterminants that C_{S,T}^{λ} is sent by ∗ (which is R-linear) to C_{T,S}^{λ}. Thus axiom (2) holds.

To show that axiom (3) holds, we use the techniques of the proof of Proposition 6.1.3. It is enough, by linearity, to check the axiom when a = C_{S′,T′}^{ν} is a standard q-codeterminant. Consider the product of the two standard q-codeterminants given by

C_{S′,T′}^{ν}C_{S,T}^{λ} = (ξ_{s′,ℓ(ν)}ξ_{ℓ(ν),t′}ξ_{s,ℓ(λ)})ξ_{ℓ(λ),t}.

Expanding the parentheses, we see from the strong straightening result that we can rewrite the whole expression as

(∑_{μ⊵λ} a_{s′′,t′′}^{μ}Y_{s′′,t′′}^{μ} ) × ξ_{ℓ(λ),t},

where the q-codeterminants Y_{s′′,t′′}^{μ} appearing are standard. This implies that if μ = λ then necessarily t′′ = ℓ(λ). We deduce that

C_{S′,T′}^{ν}C_{S.T}^{λ} ≡ a_{s′′,ℓ(λ)}^{λ}Y_{s′′,t}^{λ} (mod ⊵ λ),

where a_{s′′,ℓ(λ)}^{λ} is independent of t and T, and Y_{s′′,t}^{λ} is standard. This implies that axiom (3) holds.

Now that we know that q-codeterminants describe the structure of q-Schur algebras as cellular algebras, we can use the results in [GL] (all of which have reasonably elementary proofs) to deduce some surprisingly strong results about q-Schur algebras. For example, we can easily find a complete set of irreducible modules over a field (which was first done in [DJ3]). We can also apply the Schur functor to obtain similar results about Hecke algebras of type A. It also turns out that, under some modest extra hypotheses which q-Schur algebras satisfy, cellular algebras are quasi-hereditary. This explains why q-codeterminants provide a good context in which to study the quasi-hereditary structure of q-Schur algebras.

The following proposition establishes that the Hecke algebra H(S_{r}) can be regarded as a “cellular sub-algebra” of the q-Schur algebra, S_{q}(n, r). We assume n ≥ r, and we define ω ∈ Λ(n, r)^{+} to be (1, 1, …, 1, 0, …, 0), where there are r occurrences of the number 1. We know from [DJ3, §2] that H(S_{r}) can be identified with the subalgebra of S_{q}(n, r) given by ξ_{ω}S_{q}(n, r)ξ_{ω}.

Proposition 6.2.2 Suppose n ≥ r. Let R = ℤ[q, q^{−1}]. Let Λ = Λ^{+}(n, r), ordered by reverse dominance so that λ ≤ μ means λ ⊵ μ. Let M′(λ) be the set of strongly standard tableaux of shape λ, i.e. those standard tableaux of shape λ which contain one occurrence of each of the numbers 1, 2, …, r. Let C′ be the restriction of the map C given in Proposition 6.2.1 to the subalgebra H(S_{r}). Let ∗′ be the restriction to H(S_{r}) of the map ∗ appearing in Proposition 6.2.1. Then (Λ, M′, C′, ∗′) is a cell datum for A = H(S_{r}) over R. Proof A simple argument involving weight spaces shows that the Hecke algebra is spanned by the standard q-codeterminants it contains. It is now elementary to check that all the axioms hold.

Remark

This result is interesting because one can now recover the representation theory of the symmetric group.

One of the striking features of cellular algebras is that their representation theory is easily accessible.

Definition (Graham, Lehrer) Let A be a cellular algebra with cell datum (Λ, M, C, ∗). The left A-module W (λ) is the free R-module with basis {C_{S} : S ∈ M_{λ}} and A-action given by

a.C_{S} = ∑_{S′∈M(λ)} r_{a}(S′, S)C_{S′},

where a ∈ A and S ∈ M(λ). The module W_{λ} affords the cell representation of A corresponding to λ. Remark Axiom (3) ensures that this action is well-defined.

Proposition 6.2.3 When A = S_{q}(n, r), the modules W (λ) may be naturally identified with the q-Weyl modules of shape λ. Proof Let B be the basic tableau of shape λ. The q-Weyl module, W_{q}^{λ}, which was defined in [DJ3, Definition 3.8 (ii)], has a highest weight vector, z_{λ}. We claim that the map ρ : W_{q}^{λ} → W (λ) which sends s.z_{λ} to s.C_{B}, where s ∈ A, defines an isomorphism of A-modules.

It is enough to show that the action of standard q-codeterminants on both highest weight vectors is the same. By analysing the definition above and §5.1, we can describe this concisely.

In either case, a standard q-codeterminant of shape other than λ will annihilate either C_{B} or z_{λ}. Similar remarks apply to standard q-codeterminants of form C_{S,T}^{λ} where T ≠ B. The remaining basis elements clearly yield a basis of W (λ) when acted on C_{B}, because C_{S,B}^{λ}.C_{B} = C_{S}, and they yield a similar basis for the q-Weyl modules by Theorem 5.3.1. This shows that ρ is well-defined and completes the proof.

Graham and Lehrer [GL] introduce a certain bilinear form on W (λ) which is instrumental in describing the irreducible modules.

Proposition 6.2.4 (Graham, Lehrer) Let A be a cellular algebra with cell datum (Λ, M, C, ∗). The bilinear map ϕ_{λ} : W (λ) × W (λ) → R is defined by

C_{S_{1},T_{1}}^{λ}C_{S_{2},T_{2}}^{λ} ≡ ϕ_{λ}(C_{T_{1}}, C_{S_{2}})C_{S_{1},T_{2}}^{λ} (mod A(< λ)),

where S_{1}, T_{1}, S_{2}, T_{2} ∈ M(λ). The function ϕ_{λ} is independent of T_{1} and S_{2}, and satisfies the following equations for all x, y ∈ W (λ) and for all a ∈ A:

ϕ_{λ}(x, y) = ϕ_{λ}(y, x),  (1)
ϕ_{λ}(a^{∗}x, y) = ϕ_{λ}(x, ay).  (2)

Proof See [GL, Proposition 2.4].

We denote by Λ_{0} the subset of λ ∈ Λ satisfying ϕ_{λ} ≠ 0. We can now state Theorem 6.2.5 (Graham, Lehrer) Let A be a cellular algebra with cell datum (Λ, M, C, ∗) over a field R. Then the subspace of W (λ) given by

rad(λ) := {x ∈ W (λ) : ϕ_{λ}(x, y) = 0 for all y ∈ W (λ)}

is an A-submodule of W (λ). Define L(λ) to be W (λ)/rad(λ). Then the set {L(λ) : λ ∈ Λ_{0}} is a complete set of (representatives of equivalence classes of) absolutely irreducible A-modules. Proof See the results in [GL, §3].

Dipper and James [DJ3] define a certain bilinear form ⟨⟨ , ⟩⟩ on the q-Weyl modules. They show that it satisfies the properties (1) and (2) given in Proposition 6.2.4. We now show that the bilinear form ϕ_{λ} is essentially the same as ⟨⟨ , ⟩⟩.

Proposition 6.2.6 For each λ ∈ Λ^{+} there exists an integer m such that ϕ_{λ}(x, y) = q^{m}⟨⟨x, y⟩⟩ for all x, y ∈ W (λ). Proof We first prove that any bilinear form ⟨ , ⟩ on the q-Weyl module W (λ) satisfying properties (1) and (2) above must be a scalar multiple of the form ϕ_{λ}. Writing B as the basic tableau of shape λ, we now find that

⟨C_{S,B}^{λ}z_{λ}, C_{T,B}^{λ}z_{λ}⟩ = ⟨C_{B,T}^{λ} C_{S,B}^{λ}z_{λ}, z_{λ}⟩
= ϕ_{λ}(C_{T} , C_{S})⟨C_{B,B}^{λ}z_{λ}, z_{λ}⟩
= ϕ_{λ}(C_{T} , C_{S})⟨z_{λ}.z_{λ}⟩

The second line follows from the first by the fact that any standard q-codeterminant of shape other than λ annihilates the highest weight vector z_{λ} of the q-Weyl module. (See the proof of Proposition 6.2.3.) It was shown in [DJ3] that ⟨⟨z_{λ}, z_{λ}⟩⟩ was equal to a certain integer power of q. The result follows because ⟨⟨ , ⟩⟩ satisfies properties (1) and (2).

Examples The theory we have developed enables us to calculate explicitly the bilinear form ϕ_{λ} on the q-Weyl modules. Using Theorem 6.2.5, we can explicitly describe the irreducible modules for the q-Schur algebra, even in non-generic cases.

The Quantized Symmetric Powers

We now consider the quantized symmetric powers, in the sense of [GM]. These can be identified with the q-Weyl modules for S_{q}(n, r) in the case where λ = (r, 0, …, 0). Let C_{M} and C_{N} be standard tableaux of shape λ (i.e. row increasing tableaux with one row). Let W_{μ} be the parabolic subgroup of S_{r} which is the row stabiliser of C_{M}, and let P_{μ} be its associated Poincaré polynomial. Let P_{W} be the Poincaré polynomial of S_{r}. Using the product rule described in §2, we find that

ϕ_{λ}(C_{M}, C_{N}) = δ_{M,N} \frac{P_{W}}{P_{μ}} (q).

Note that this quotient of polynomials becomes a multinomial coefficient as q is replaced by 1 (compare with [M, p. 83, Example 1]).

The Quantized Exterior Powers

Suppose now that n ≥ r. We consider the quantized exterior powers, which can be identified with the q-Weyl modules for S_{q}(n, r) in the case where λ = ω. Let C_{M} and C_{N} be standard tableaux of shape λ (i.e. strictly column increasing tableaux with one column). Using the product rule described in §2, we find that

ϕ_{λ}(C_{M}, C_{N}) = δ_{M,N}.

This implies that the quantized exterior power is always absolutely irreducible when q is a unit.

One can obtain other interesting results for q-Schur algebras by using the theory of cellular algebras. Other results contained in [GL] include criteria for cellular algebras to be semisimple and quasi-hereditary.

As one would expect from corresponding results in the classical case (see [G2]) and the preceding discussion of cellular algebras, q-codeterminants are well suited to describe the structure of the q-Schur algebra as a quasi-hereditary algebra, in the sense of Cline, Parshall and Scott. With any quasi-hereditary algebra one associates a defining sequence, which is a certain chain of two-sided ideals of the algebra satisfying various technical conditions. (For the definitions and basic properties of quasi-hereditary algebras, the reader is referred to [M, §3.3].)

We recall the defining sequence for S_{q}(n, r): Theorem 6.2.7 (Parshall, Wang)

Let e_{i} be the idempotent of S_{q}(n, r) given by ξ_{λ^{1}} + ⋯ + ξ_{λ^{i}}, and define J_{i} to be the ideal Se_{i}S, where S = S_{q}(n, r) and J_{0} is taken to be 0. Then S_{q}(n, r) is quasi-hereditary with weight poset Λ^{+} and

{0} = J_{0} ⊆ J_{1} ⊆ ⋯ ⊆ J_{t} = S

is a defining sequence for S. Proof See the remark in [Do3, §4].

We can restate this result in terms of q-codeterminants, thus obtaining a free ℤ[q, q^{−1}]-basis of each J_{i}. Corollary 6.2.8 The ideal J_{i} has as a free ℤ[q, q^{−1}]-basis the set of all standard q-codeterminants of shapes λ^{1}, …, λ^{i}, and is equal to the ideal Y (π) corresponding to the increasing saturation π = {λ^{1}, …, λ^{i}}. Proof Since J_{i} is generated by e_{i}, we see it contains ξ_{λ^{s}} = ξ_{λ^{s}}e_{i}ξ_{λ^{s}} where s ≤ i. It is therefore generated as a two-sided ideal by the idempotents ξ_{λ^{s}} where s ≤ i. By considering the natural basis of S_{q}(n, r) we now find that J has as a spanning set the set of elements ξ_{a,b}ξ_{λ^{s}}ξ_{c,d} where s ≤ i, or, more economically, the set ξ_{a,ℓ(λ^{s})}ξ_{ℓ(λ^{s}),b} where again s ≤ i. This is precisely the set of q-codeterminants of shape λ^{s}. Because ≻ refines ▷, the set {λ^{1}, …, λ^{s}}, where s ≤ i, is an increasing saturation, and the result now follows from Proposition 6.1.4.

Before proceeding, it is worth noting two similarities between the basis of standard quantized codeterminants and Du’s canonical (intersection cohomology) basis {θ_{λ,μ}^{d}}. This basis, which was introduced in [D2, §2], is in natural bijection with Dipper and James’s basis {ϕ_{λ,μ}^{d}}. Like the basis of standard q-codeterminants, it has the property that bases for highest weight modules can be found by acting all the basis elements on the highest weight vector and taking the nonzero results (see [D3, Theorem 5.3] and Theorem 5.3.1). It is shown in [D3, Theorem 3.3] that the canonical basis also gives rise to the defining sequence for S_{q}(n, r) in a natural way; again, each J_{i} is spanned by the basis elements it contains.

Du in [D1, Theorem 3.3] obtained (explicitly) the following result on quotients of Schur algebras, which is a special case of Theorem 6.3.5. Proposition 6.2.9 (Du)

Let

0 = J_{0} ⊆ J_{1} ⊆ ⋯ ⊆ J_{t} = S(2, r)

be the defining sequence for the classical Schur algebra S(2, r). Then

\frac{S(2, r)}{J_{i}} ≅ S(2, r − 2i).

Proof See [D1, Theorem 3.3].

We wish to find an analogue of this result in the quantized case for arbitrary n. It turns out that in general, quotienting out by ideals in the defining sequence does not produce smaller q-Schur algebras; the case n = 2 is a simple case because the order ⊵ happens to be total. It is however true that by quotienting by a suitable ideal corresponding to an increasing saturation, one can find an analogue of Proposition 6.2.9.

6.3. Epimorphisms between v-Schur algebras

It is convenient at this point to switch notation and to start working with the v-Schur algebra S_{v}(n, r) and its associated v-codeterminants Ŷ_{i,j}^{λ} as defined in §4.2. We write Ŷ(π) for the ideal spanned by all v-codeterminants with shapes lying in π. Recall that S_{v}(n, r) and S_{q}(n, r) are related as rings via

S_{v}(n, r) ≅ A ⊗ S_{q}(n, r)

and as ℤ[q, q^{−1}]-modules via

S_{v}(n, r) ≅ S_{q}(n, r) ⊕ v.S_{q}(n, r).

We now aim to describe an epimorphism between certain v-Schur algebras and to investigate its properties.

In the main proof, we will need certain results in the representation theory of quantized enveloping algebras. We deal with these now. Definition Define L ∈ V^{⊗n} to be the quantized exterior power (in the sense of §5.2) given by

∑_{σ∈S_{n}}(−v)^{−ℓ(σ)}e_{σ(1)} ⊗ ⋯ ⊗ e_{σ(n)}.

Lemma 6.3.1 The vector L spans a 1-dimensional U module affording the representation which sends E_{i} and F_{i} to zero for all i, and K_{j} to v.1 for all j. Proof The actions of E_{i} and F_{i} on L can be deduced from Lemma 5.2.7. The assertion for the K_{j} is clear by direct consideration of its action.

Lemma 6.3.2 Let b ∈ V^{⊗r}. Then the action of U on V^{⊗(r+n)} has the following properties.

1. If u ∈ U^{+} then u.(L ⊗ b) = L ⊗ (u.b).

2. If u ∈ U^{−} then u.(L ⊗ b) = L ⊗ (u.b).

3. If u is of the form

\left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] ,

where ∑_{i=1}^{n} t_{i} = n + r, then u.(L ⊗ b) = L ⊗ (u′.b), where u′ is given by

\left[{K_{1}; 0 \atop t_{1} − 1}\right] ⋯ \left[{K_{n}; 0 \atop t_{n} − 1}\right] ,

or 0 if this does not make sense (because one or more of the t_{i} was zero). Proof We first deal with the proof of part 1. We see from the nature of the identity that it is enough to check the statement for the algebra generators of U^{+}, i.e. the identity element, 1, and the elements E_{i}. The statement is clear for u = 1. If u = E_{i}, we find that u acts (via the coassociative comultiplication Δ) on L ⊗ b as

E_{i} ⊗ K_{i}K_{i+1}^{−1} + 1 ⊗ E_{i}.

The proof of part 1 now follows, because E_{i} annihilates L, as shown in Lemma 6.3.1.

The proof of part 2 follows similar lines to the proof of part 1. In this case, the action of F_{i} on L ⊗ b is given by acting

K_{i}^{−1}K_{i+1} ⊗ F_{i} + F_{i} ⊗ 1.

Again, we find that F_{i} annihilates L, so only the first term has any effect. One easily checks that K_{i}^{−1}K_{i+1} acts as 1 on L, and part 2 now follows.

The proof of part 3 follows from Theorem 2.2.7 and Corollary 4.1.10. We see from those results that u will fix any basis vector of form e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{s}} ∈ V^{⊗(r+n)} where the multiplicity of e_{j} in the tensor is equal to t_{j}. Basis vectors not of this form are annihilated by u. The result follows because the multiplicity of each e_{j} in the element L is equal to 1.

The following basic fact about standard tableaux turns out to be important in Theorem 6.3.5. Lemma 6.3.3 Denote by T (n, r) the set of standard tableaux with r boxes and entries from {1, …, n}. There is an injection ι from T (n, r) to T (n, r + n) given by adding an extra column of length n and with entries 1, 2, …, n to the left of the tableau. Proof This is immediate, because in a standard tableau, all entries in row i must be greater than or equal to i. This implies that the addition of the new column does not affect the property of standardness. It is clear that the new tableau is an element of T (n, r + n). Example Suppose n = 3 and r = 4. Let T be the tableau given by

1 3 2 2 .

Then ι(T) is given by

1 2 3 1 3 2 2 .

Lemma 6.3.4 Let [A][A′] be a standard v-codeterminant corresponding to a pair of standard tableaux ⟨T, T′⟩. Then

A_{11} ≥ A_{22} ≥ ⋯ ≥ A_{nn} = A′_{nn} ≤ ⋯ ≤ A′_{22} ≤ A′_{11}.

Proof Note A_{ii} is the number of entries equal to i in row i of the tableau T; these entries must therefore occur at the left end of the row. Similarly, A′_{ii} is the number of entries equal to i in row i of T′. For any standard tableau, one sees that the number of entries equal to i in row i is greater than or equal to the number of entries equal to i + 1 in row i + 1. This implies that

A_{11} ≥ ⋯ ≥ A_{nn}

and

A′_{11} ≥ ⋯ ≥ A′_{nn}.

The fact that A_{nn} = A′_{nn} follows from the requirement that T and T′ must be of the same shape. We now find that A_{nn} and A′_{nn} are both expressions for the number of columns of length n in T (or T′). (This is because all entries in the n-th row are equal to n.)

Definition

We define the linear map ψ_{n,r} : S_{v}(n, r + n) → S_{v}(n, r) by its effects on the standard vcodeterminants as follows:

ψ_{n,r}([A][A′]) = \{ \begin{matrix}[A − I][A′ − I] & if A_{nn} ≥ 1, \\ 0 & otherwise. \end{matrix}

Remarks

We see from Lemma 6.3.4 that this definition makes sense, and we see from Lemma 6.3.3 that it takes standard v-codeterminants in S_{v}(n, r + n) to zero or to standard v-codeterminants in S_{v}(n, r). Subtraction of the identity matrix (if possible) from each component of the standard v-codeterminant corresponds to removing the first column from each member of the corresponding pair of tableaux (if it is a column of length n).

Example Denote by [T, T′] the standard v-codeterminant corresponding to the ordered pair of standard tableaux ⟨T, T′⟩. Then

[ 1 2 3 1 2 2 , 1 2 3 2 3 3 ] ⟶^{ψ_{3,3}} [ 1 2 2 , 2 3 3 ]

and

[ 1 2 2 3 3 3 , 1 2 2 3 2 2 ] ⟶^{ψ_{3,3}} 0.

We are now ready to state and prove an epimorphism theorem. Theorem 6.3.5

1. The map ψ_{n,r} is an epimorphism of ℤ[v, v^{−1}]-algebras. Its kernel is the ideal Ŷ(π), where π is the set of λ ∈ Λ^{+} such that a tableau of shape λ has no column of length n.

2. For a natural number k, define the linear map ψ_{n,r,k} : S_{v}(n, r + kn) → S_{v}(n, r) by

ψ_{n,r,k}([A][A′]) = \{ \begin{matrix}[A − kI][A′ − kI] & if A_{nn} ≥ k, \\ 0 & otherwise. \end{matrix}

(Note that ψ_{n,r} = ψ_{n,r,1}.)

The map ψ_{n,r,k} is an epimorphism of algebras whose kernel is the ideal Ŷ(π_{k}), where π_{k} is the set of λ ∈ Λ^{+} such that a tableau of shape λ has strictly fewer than k columns of length n.

3. The restriction of ψ_{n,r,k} to S_{v}^{+}(n, r+kn) (the Borel subalgebra of S_{v}(n, r+kn)) has the following

effect on the basis elements [A] of S_{v}(n, r + kn):

ψ_{n,r,k}([A]) = \{ \begin{matrix}[A − kI] & if there does not exist i such that A_{ii} < k, \\ 0 & otherwise. \end{matrix}

The image of the restriction of ψ_{n,r,k} is precisely S_{v}^{+}(n, r), and ker ψ_{n,r,k} is spanned by the basis elements [A] which it contains. There is an analogous result for the Borel subalgebra S^{−}.

Proof

We first prove that ψ is an epimorphism. Surjectivity comes from Lemma 6.3.3, because the v-co-determinant in S_{v}(n, r) corresponding to ⟨T, T′⟩ is the image under ψ of the v-codeterminant in S_{v}(n, r + n) corresponding to ⟨ι(T), ι(T′)⟩.

Next we define a certain map ψ from S_{v}(n, r + n) to S_{v}(n, r) which will turn out to coincide with ψ_{n,r}. Let x ∈ S_{v}(n, r + n), and choose u ∈ U such that θ_{r+n}(u) = x for the homomorphism θ_{r+n} : U(gl_{n}) → S_{v}(n, r + n). (This means that u and x act in the same way on V^{⊗(r+n)}, whichever u is chosen.) We now embed tensor space V^{⊗r} in tensor space V^{⊗(r+n)} via b ↦ L ⊗ b. The Hecke algebra H_{r} which acts on V^{⊗r} on the right may be naturally identified with a subalgebra of the Hecke algebra H_{r+n} acting on V^{⊗(r+n)} on the right by sending the generator T_{(p,p+1)} to the generator T_{(p+n,p+n+1)}.

We define the linear map ψ to send x ∈ S_{v}(n, r + n) to the endomorphism X of V^{⊗r} given by u.(L ⊗ b) = L ⊗ X(b), where u is as above. The existence of X is a consequence of Lemma 6.3.2, as we shall explain shortly, and X is unique (i.e. independent of which u is chosen) because S_{v}(n, r +n) is the faithful quotient of the action of U on V^{⊗(r+n)}. It was shown in Theorem 4.2.5 that there exists such a u with the property that u = u^{−}u^{0}u^{+}, where u^{−} ∈ U^{−}, u^{+} ∈ U^{+} and u^{0} is of form

\left[{K_{1}; 0 \atop t_{1}}\right] ⋯ \left[{K_{n}; 0 \atop t_{n}}\right] ,

where ∑_{i=1}^{n} t_{i} = r + n. It now follows from Lemma 6.3.2 that X exists, since X(b) = u^{−}u′u^{+}.b, where u^{−} and u^{+} are as before, and u′ is the element of U^{0} given by

\left[{K_{1}; 0 \atop t_{1} − 1}\right] ⋯ \left[{K_{n}; 0 \atop t_{n} − 1}\right] ,

or zero if this does not make sense. The embedding of H_{r} in H_{r+n} which we describe implies that the endomorphism of V^{⊗r} given by X commutes with the action of the associated Hecke algebra, and hence, by the double centralizer theorem, is equal to an element of S_{v}(n, r). Thus ψ is well-defined.

To show that ψ = ψ_{n,r}, we recall Theorem 4.2.5. Choose [A][A′] to be a standard codeterminant of S_{v}(n, r+n). We can rewrite this as [A][D][A′] for a suitable diagonal matrix D. (The (i, i)-entry of D is λ_{i}, where λ is the shape of the standard v-codeterminant concerned.) The Theorem exhibits an element u of the form we want which is naturally expressible as u^{−}u^{0}u^{+}. The element u^{−} determines the entries below the diagonal of A (in a way described in Proposition 4.1.7), u^{+} determines the entries above the diagonal of A′, and

u^{0} = \left[{K_{1}; 0 \atop λ_{1}}\right] ⋯ \left[{K_{n}; 0 \atop λ_{n}}\right] .

Now consider the action of u^{−}u^{0}u^{+} on L ⊗ b, where b ∈ V^{⊗r}. We can now deduce several results from Lemma 6.3.2 and the expressions for θ_{r+n}(u^{−}) and θ_{r+n}(u^{+}) (given in Propositions 4.1.7 and 4.1.3):

a) The element u^{0} acts as [D − I] (or 0 if this does not make sense).

b) The element u^{−}u^{0} acts as [A − I] (or 0 if this does not make sense).

c) The element u^{0}u^{+} acts as [A′ − I] (or 0 if this does not make sense). We know from the proof of Theorem 4.2.5 that u acts in the same way on V^{⊗(r+n)} as u^{−}u^{0}u^{0}u^{+},

so this implies that u acts as [A − I][A′ − I], or 0 this does not make sense. Hence the maps ψ and ψ_{n,r} agree, and ψ_{n,r} is an algebra homomorphism (because ψ clearly is, being defined in terms of an endomorphism algebra).

We postpone proof of the description of ker(ψ_{n,r}) because it is a special case of ker(ψ_{n,r,k}) for k = 1.

The proof that ψ_{n,r,k} is an epimorphism of algebras follows similar lines to the corresponding proof in part 1, except that the embedding of V^{⊗r} in V^{⊗(r+kn)} is via

b ↦ \underbrace{L ⊗ ⋯ ⊗ L}_{k} ⊗b.

It was explained earlier why A_{nn} is the number of columns in the tableau of the same shape as the v-codeterminant, and because ψ_{n,r,k} takes standard v-codeterminants to zero or to standard v-codeterminants, we see that ker ψ_{n,r,k} is as claimed.

We can prove part 3 by using the facts labelled above as a), b) and c). We concentrate on the case of S_{v}^{+}(n, r + kn) because the case of S_{v}^{−}(n, r + kn) is almost exactly the same. Recall that S^{+} has as a basis all basis elements [U] of S_{v}(n, r + kn) such that U is an upper triangular matrix, and similarly S^{−} has as a basis all lower triangular basis elements [L]. We can now check from the preceding arguments that ψ_{n,r,k}[A] is as claimed, when [A] is lower or upper triangular. Note that ψ_{n,r,k} takes basis elements of S_{v}^{+}(n, r + kn) to zero or to basis elements of S_{v}^{+}(n, r), and that all triangular basis elements of S_{v}^{+}(n, r) turn up in this way. This shows that the restricted image and restricted kernel of ψ_{n,r,k} are as claimed, completing the proof of part 3.

Finally we prove that π and π_{k} are increasing saturations. Since π = π_{1}, we concentrate on the case for general k. Suppose a tableau of shape λ has fewer than k columns, and μ ▷ λ. We need to show that μ has fewer than k columns. Saying that λ has fewer than k columns is equivalent to saying that λ_{n} < k. Since μ ▷ λ, we find that in particular,

λ_{1} + ⋯ + λ_{n−1} ≤ μ_{1} + ⋯ + μ_{n−1},

implying that μ_{n} ≤ λ_{n} < k as required, since ∑_{i=1}^{n} λ_{i} = ∑_{i=1}^{n} μ_{i} = r.

This completes the proof.

Remarks Donkin [Do1] proves an analogue of this result in an even more general setting—that of generalised Schur algebras, although the analogue of the map ψ is not found explicitly. The map ψ_{n,r} turns out [Do2] to be the dual of the coalgebra injection from A(n, r) to A(n, r + n) given by multiplication by the determinant function. (Here, S(n, r) is the dual of A(n, r).)

Notice that in the proof of Theorem 6.3.5, the natural basis for the Borel subalgebras plays the same rôle as the v-codeterminant basis for the v-Schur algebras. It should however be noted that the Borel subalgebras considered are not generalised Schur algebras [Do2].

6.4 Cellular inverse limits

Using the epimorphisms between v-Schur algebras studied in §6.3, we can define certain inverse limits of v-Schur algebras. It will turn out that the inverse limit has a natural cellular structure, and the quantized enveloping algebra U(sl_{n}) embeds in it in a natural way.

First we note that the v-Schur algebra can be given a cellular structure which differs slightly from the one in §6.2.

Proposition 6.4.1 Let Λ, M and ≤ be as in Proposition 6.2.1. Let R = A. Let C be the map taking the pair of elements ⟨S, T ⟩ (where S, T ∈ M(λ)) to the associated standard v-codeterminant. Let ∗ be the anti-automorphism of S_{v}(n, r) sending [A] to [A^{⊤}] (i.e. transposition of basis elements). Then (Λ, M, C, ∗) is a cell datum for A = S_{v}(n, r) over R. Proof The fact that ∗ is an anti-automorphism was proved in [BLM, Lemma 3.10]. The rest of the proof is analogous to the proof of Proposition 6.2.1.

Definitions

It will be convenient to introduce the v-Schur algebra S_{v}(n, r) for r = 0. This is nothing other than the base ring A, regarded as a quotient of S_{v}(n, n) via the map ψ_{n,0} : S_{v}(n, n) → S_{v}(n, 0). This map takes the standard v-codeterminant [I][I] (where I is the identity matrix) to 1 ∈ A, and other standard v-codeterminants to 0.

Suppose 0 ≤ r < n (where r is an integer as usual). We define Ŝ_{v}(n, r) to be the set of semi-infinite sequences (x_{i}) satisfying

⋯ ⟶^{ψ_{n,r+3n}} x_{3} ⟶^{ψ_{n,r+2n}} x_{2} ⟶^{ψ_{n,r+n}} x_{1} ⟶^{ψ_{n,r}} x_{0},

where the element x_{k} lies in S_{v}(n, r + kn).

Suppose r is as above. Define Λ̂(n, r) to be the set

{λ ∈ Λ^{+}(n, r + kn) for some k ∈ ℕ with λ_{n} = 0}.

Lemma 6.4.2

Let λ lie in Λ^{+}(n, r + kn) for some k ≥ 0. Denote by ω the n-tuple (1, 1, …, 1). Then λ is of form λ′ + λ_{n}ω, where λ′ ∈ Λ̂. Any standard v-codeterminant [A][A′] ∈ S_{v}(n, r + kn) of shape λ satisfies

ψ_{n,r+(k−λ_{n})n,λ_{n}}([A][A′]) = [A − λ_{n}I][A′ − λ_{n}I],

where the shape of [A − λ_{n}I][A′ − λ_{n}I] lies in Λ̂(n, r).

If T, T′ are elements of the same shape in Λ̂(n, r) ∩ Λ^{+}(n, r + (k + 1)n) then C(⟨T, T′⟩) is a standard v-codeterminant of S_{v}(n, r + (k + 1)n) which is mapped to zero by ψ_{n,r+kn}. Proof All the assertions are immediate from the definitions.

Example Let r = 0 and n = 3. Let

T = 1 2 3 1 2 2

and

T′ = 1 2 3 2 3 3 .

Then T and T′ are both of shape λ = (3, 2, 1) which lies in Λ^{+}(3, 6). This is of form λ′ + ω, where λ′ = (2, 1, 0). One easily sees that

ψ_{3,3,1}(C(⟨T, T′⟩))

is a standard v-codeterminant of shape λ′ ∈ Λ̂(3, 0) ∩ Λ^{+}(3, 3) which is mapped to zero by ψ_{3,0}.

We introduce a partial order, ≤, on the elements of Λ̂(n, r) which is analogous to the dominance order on Λ^{+}.

Definition Let λ and μ be elements of Λ̂(n, r). We define will write λ ≤ μ if

∑_{i=1}^{n} λ_{i} = ∑_{i=1}^{n} μ_{i} + c.n

for some nonnegative integer c, and λ ⊵ μ + c.ω.

Lemma 6.4.3 The order ≤ is a partial order on Λ̂(n, r). Proof This will follow from the fact that ⊵ is a partial order, and the observation that λ ⊵ μ if and only if λ + ω ⊵ μ + ω for λ, μ ∈ Λ^{+}(n, r).

Reflexivity is clear from the definition of ≤.

Let λ, μ ∈ Λ̂(n, r). Suppose λ ≤ μ and μ ≤ λ. The definition of ≤ forces c = 0 in this case, and we have both λ ⊵ μ and λ ⊴ μ. Since ⊵ is a partial order, we find that λ = μ. This proves antisymmetry.

Suppose λ, μ, ν ∈ Λ̂(n, r), λ ≤ μ and μ ≤ ν. We see from the definitions that

λ ⊵ μ + c_{1}.ω

and

μ ⊵ ν + c_{2}.ω

for some natural numbers c_{1} and c_{2}. Using the remark at the beginning of the proof, we find that

μ + c_{1}ω ⊵ ν + (c_{1} + c_{2})ω,

and thus λ ≤ ν by definition of ≤ and by transitivity of ⊵. Thus ≤ is transitive.

This proves that ≤ is a partial order.

One can naturally define a set of standard v-codeterminants for Ŝ_{v}(n, r) analogous to the standard v-codeterminants for the v-Schur algebras. If T and T′ are standard tableaux of the same shape, we will write C(T, T′) for the associated standard v-codeterminant, using the map C which appears in Proposition 6.4.1. It should be noted that this set of v-codeterminants will not be a basis of Ŝ_{v}(n, r): the set is linearly independent, but is not a spanning set.

Definition Let C(T, T′) be a standard v-codeterminant of S_{v}(n, r + kn) of shape λ ∈ Λ̂(n, r). Define the sequence (x_{i}) (where x_{i} ∈ S_{v}(n, r + in)) via

x_{i} := \{ \begin{matrix}0 & if i < k, \\ C(T, T′) & if i = k, \\ C(ι^{i−k}(T), ι^{i−k}(T′)) & if i > k. \end{matrix}

Here, the injection ι is as in §6.3. It follows from Theorem 6.3.5 and Lemma 6.4.2 that the sequence (x_{i}) defines an element of Ŝ_{v}(n, r). We will denote this element by Ĉ(T, T′). Such elements will be referred to as standard v-codeterminants of Ŝ_{v}(n, r).

In the results which follow, we will be considering certain “infinite sums”

∑_{⟨T,T′⟩} â(T, T′)Ĉ(T, T′).

Here, the elements â(T, T′) lie in A, but there may be infinitely many v-codeterminants C(T, T′) occurring with nonzero coefficient. It to be understood that â(T, T′) is the total coefficient of Ĉ(T, T′) in the sum. (“Repeats” are not allowed.) This infinite sum is interpreted as an element of Ŝ_{v}(n, r) as follows.

Consider a typical sum

∑_{⟨T,T′⟩} â(T, T′) Ĉ(T, T′),

where the sum is infinite (as explained above) and the codeterminants appearing are standard. This is a sequence (x_{i}) (where x_{i} ∈ S_{v}(n, r+in) as usual) as follows. With respect to the basis of standard v-codeterminants of S_{v}(n, r + kn), the coefficient a(S, S′) of C(S, S′) in x_{k} is given by

a(S, S′) = \{ \begin{matrix}â(T, T′) & if (ι^{m}(T), ι^{m}(T′)) = (S, S′) for a nonnegative integer m, \\ 0 & otherwise. \end{matrix}

Note that the integer m introduced above must be unique if it exists. Lemma 6.4.2 shows that this sequence (x_{i}) is indeed an element of Ŝ_{v}(n, r).

Proposition 6.4.4 Each element of Ŝ_{v}(n, r) can be expressed uniquely in the form

∑_{⟨T,T′⟩} â(T, T′)Ĉ(T, T′),

where the elements â(T, T′) lie in A and the v-codeterminants Ĉ(T, T′) appearing are standard. Proof Suppose we have a sequence (x_{i}) giving an element of Ŝ_{v}(n, r). We define the coefficient â(T, T′) of a standard v-codeterminant Ĉ(T, T′) to be the coefficient a(T, T′) of C(T, T′) in x_{k}, where k is such that the tableaux T and T′ have r + kn boxes. Then we have

(x_{i}) = ∑_{⟨T,T′⟩} â(T, T′)Ĉ(T, T′).

Thus any element of Ŝ_{v}(n, r) is expressible in terms of an infinite sum of the required form. Moreover this expression is unique, using the above definition of the infinite sum.

Lemma 6.4.5 The A-module Ŝ_{v}(n, r) is an associative algebra in the obvious way, i.e. if x, y ∈ Ŝ_{v}(n, r) and x = (x_{i}), y = (y_{i}) then xy := (x_{i}y_{i}). Proof This works because the maps ψ are algebra homomorphisms.

Note For our purposes, it is necessary to alter the axioms for a cellular algebra slightly. (For finite-dimensional algebras, this alteration makes no difference at all.) We now allow infinite R-linear combinations of basis elements C_{S,T}^{λ} in the axioms. This alters the three axioms as follows.

1. We drop the hypothesis that the image of C is an R-basis for A. Instead, we ensure that each element of A can be written uniquely in the form

∑ a(S, T) C(S, T),

where a(S, T) ∈ R is the coefficient of C(S, T) in the sum. The sum may contain infinitely many elements C(S, T) which occur with nonzero coefficients.

2. We strengthen the hypothesis that the anti-automorphism ∗ is R-linear by requiring it to satisfy

∗ : ∑ a(S, T) C(S, T) ↦ ∑ a(S, T) C(T, S),

where the sums satisfy the same rules as above.

3. We extend the R-submodule A(< λ) to consist of all elements

∑ a_{S′′,T′′}C_{S′′,T′′}^{μ}

where μ < λ, S′′ ∈ M(μ) and T′′ ∈ M(μ). The scalar a_{S′′,T′′} ∈ R is the coefficient of C_{S′′,T′′}^{μ} in the sum. Again, infinitely many distinct C_{S′′,T′′}^{μ} may appear.

We call an algebra satisfying these modified axioms a generalised cellular algebra, and we call the quadruple (Λ, M, C, ∗) a generalised cell datum for the algebra. Note that a generalised cellular algebra A over R satisfying the additional condition that the image of C is an R-basis for A is a cellular algebra in the usual sense. Also note that finite-dimensional generalised cellular algebras are the same as finite-dimensional cellular algebras.

Remarkably, many of the properties of cellular algebras studied in [GL] carry over to generalised cellular algebras, and the proofs are the same. For example, the finite-dimensional cell modules W (λ) of §6.2 may be defined in the same way for generalised cellular algebras, and they have the properties one would expect.

We can now give a generalised cell datum for Ŝ_{v}(n, r).

Proposition 6.4.6

Let R = A. For λ ∈ Λ̂(n, r), let M̂(λ) = M(λ), where M is as in Proposition 6.2.1. Let Ĉ be as above. Define the A-linear map ˆ∗: Ŝ_{v}(n, r) ↦ Ŝ_{v}(n, r) by

ˆ∗: ∑ â(T, T′) Ĉ(T, T′) ↦ â(T, T′) Ĉ(T′, T).

Then (Λ̂, M̂, Ĉ, ˆ∗) is a generalised cell datum for Ŝ_{v}(n, r), where Λ̂ is partially ordered by ≤. Proof Axiom (1) for generalised cellular algebras follows from Proposition 6.4.4 and the observation that M̂(λ) is a finite set for each λ.

We must show that the map ˆ∗ maps Ŝ_{v}(n, r) to itself. Let (x_{i}) be an element of Ŝ_{v}(n, r). Then (x_{∗} ; i) is another such sequence, because the maps ψ commute with the anti-automorphisms ∗ of the v-Schur algebras. (This is easy to see from Theorem 6.3.5.) The fact that it is an anti-automorphism follows from the definition of multiplication in Ŝ_{v}(n, r). Axiom (2) now follows.

To check axiom (3), we choose a standard v-codeterminant, Ĉ_{T,T′}^{λ}. Define k ∈ ℕ by stating that

∑_{i=1}^{n} λ_{i} = r + kn,

where 0 ≤ r < n is as in the statement of the proposition. Pick any element x ∈ Ŝ_{v}(n, r). Suppose that Ĉ_{S,S′}^{μ} occurs with coefficient r_{x} ≠ 0 in the expression for x.Ĉ_{T,T′}^{λ}. We know from Lemma 6.4.2 that Ĉ_{T,T′}^{λ} is a sequence (y_{i}) where y_{i} = 0 if i < k. Because of the way the multiplication is defined, it must be the case that the sequence (z_{i}) corresponding to Ĉ_{S,S′}^{μ} also satisfies z_{i} = 0 for i < k. This implies that

∑_{i=1}^{n} μ_{i} = r + k′n

for some k′ ≥ k. This means that if μ ⊵ λ + (k′ − k).ω, we will have μ ≤ λ.

Consider the expression

x_{k′}.C(ι^{k′−k}(T), ι^{k′−k}(T′))

in S_{v}(n, r + k′n). Theorem 6.3.5 shows that the coefficient of C(S, S′) in this expression is r_{x}. Proposition 6.4.1 now shows that μ ⊵ λ + (k′ − k).ω, and thus μ ≤ λ.

If C(S, S′) is of shape λ, the above technique and Proposition 6.4.1 show that r_{x} is independent of T′. (Note that k = k′ in this situation.)

Axiom (3) now follows.

It is interesting to note that Ŝ_{v}(n, r) has a natural topology on it which is compatible with the cellular structure.

Definitions For each nonnegative integer k, let B_{k,r} be the subset of Ŝ_{v}(n, r) given by

{(x_{i}) : i < k ⇒ x_{i} = 0}.

We stipulate that the set {B_{k,r} : k ∈ ℕ} is a base of neighbourhoods of 0 in Ŝ_{v}(n, r).

Proposition 6.4.7

(i) The algebra Ŝ_{v}(n, r) has the structure of a Hausdorff topological ring in which the operation ˆ∗ is a homeomorphism. The subsets B_{k,r} are ideals of Ŝ_{v}(n, r). With respect to this topology, Ŝ_{v}(n, r) is complete.

(ii) As algebras,

\frac{Ŝ_{v}(n, r)}{B_{k+1,r}} ≅ S_{v}(n, r + kn).

Proof We first prove part (i). The operation + is clearly continuous by construction of the topology from the base. It is easy to check that the subsets B_{k,r} are ideals of Ŝ_{v}(n, r), which means that negation and multiplication are continuous. Clearly, ˆ∗ maps B_{k,r} to B_{k,r}, so it is continuous too (and hence a homeomorphism, since it is self-inverse). The Hausdorff property follows from the observation that

⋂_{i=0}^{∞} B_{k,r} = {0}.

It is now not hard to see that every Cauchy sequence will converge, and thus that Ŝ_{v}(n, r) is complete.

We now prove part (ii). The A-linear map which projects the element (x_{i}) of Ŝ_{v}(n, r) to x_{k} is an algebra homomorphism with kernel B_{k+1,r}. The proof follows.

We now study the direct sum of all the algebras Ŝ_{v}(n, r) for different values of r.

Definitions We define

Ŝ_{v}(n) := ⨁_{r=0}^{n−1} Ŝ_{v}(n, r).

This direct sum is a direct sum of A-algebras. A base of neighbourhoods of zero is the set of subsets B_{k} as k ranges through ℕ and

B_{k} := ⨁_{r=0}^{n−1} B_{k,r}.

Proposition 6.4.8 Let R = A. Let Λ^{∘} = Λ^{∘}(n) be the disjoint union ⋃_{r=0}^{n−1} Λ̂(n, r). (Note that Λ^{∘} is the set of all partitions of natural numbers into at most n − 1 pieces.) We order Λ^{∘} via ≤^{∘}. For λ, μ ∈ Λ^{∘}, we say λ ≤^{∘} μ if λ and μ both lie in Λ̂(n, r) for some r and λ ≤ μ with respect to the order ≤ on Λ̂(n, r).

For each λ ∈ Λ^{∘}, let M^{∘}(λ) be the set of standard tableaux of shape λ. Let C^{∘} be the map taking a pair of standard tableaux ⟨T, T′⟩ (of the same shape λ ∈ Λ^{∘}) to the element of Ŝ_{v}(n) corresponding to the standard v-codeterminant Ĉ_{T,T′}^{λ}. Let ∗^{∘} be the anti-automorphism satisfying

∗^{∘} : ∑ a^{∘}(T, T′) C^{∘}(T, T′) ↦ ∑ a^{∘}(T, T′) C^{∘}(T′, T).
Then (Λ^{∘}, M^{∘}, C^{∘}, ∗^{∘}) is a generalised cell datum for Ŝ_{v}(n).

Proof This is relatively easy to show after the work done in Proposition 6.4.6.

To verify axiom (1), we remark that ≤^{∘} is a partial order and M^{∘}(λ) is a finite set for each λ. The other assertions of the axiom follow from the direct sum decomposition in the definition of Ŝ_{v}(n).

Axiom (2) is clear from the proof of Proposition 6.4.6 and the direct sum decomposition. Axiom (3) follows quickly from the definition of ≤^{∘} and the direct sum decomposition.

Proposition 6.4.9 The algebra Ŝ_{v}(n) is a Hausdorff topological ring, where the operation ∗^{∘} is continuous. The subsets B_{k} are ideals of Ŝ_{v}(n). With respect to this topology, Ŝ_{v}(n) is complete. Proof The proof is very similar to the proof of Proposition 6.4.7 (i).

The motivation behind studying Ŝ_{v}(n) is that we now show how to embed the quantized enveloping algebra U(sl_{n}) in it. To do this, we will first work over the field ℚ(v) and embed U(sl_{n}) in Ŝ′(n) := ℚ(v) ⊗ Ŝ_{v}(n).

Following [D4], we identify U(sl_{n}) with the subalgebra of U(gl_{n}) generated by the elements

E_{i}, F_{i}, K_{i}K_{i+1}^{−1}, K_{i}^{−1}K_{i+1},

as 1 ≤ i < n.

Definitions Let Ŝ′(n) be as above. We write Ŝ′(n, r) for ℚ(v) ⊗ Ŝ_{v}(n, r).

Let u ∈ U(sl_{n}), and let 0 ≤ r < n.

The map θ̂_{r} : U → Ŝ′(n, r) is defined to take u to the sequence (θ_{r+in}(u)), where as usual θ_{r+in} is a certain epimorphism from U(gl_{n}) to S_{v}(n, r + in). (It remains to be seen that the resulting sequence is an element of Ŝ′(n, r).)

The map θ̂: U → Ŝ′(n) is defined to take u to

⨁_{r=0}^{n−1} θ̂_{r}(u).

Lemma 6.4.10

(i) For each u ∈ U(sl_{n}) the sequence (θ_{r+in}(u)) is an element of Ŝ′(n, r). (ii) The map θ̂_{r} : U → Ŝ′(n, r) defined by θ̂_{r}(u) = (θ_{r+in}(u)) is an algebra homomorphism. (iii) The map θ̂: U → Ŝ′(n) defined by θ̂(u) = ⨁_{r=0}^{n−1} θ̂_{r}(u) is an algebra homomorphism. Proof It is clear that (ii) and (iii) will hold if the codomains are as claimed in the definitions, because all the maps θ_{m} are homomorphisms. Thus, the only nontrivial part of the proof is checking that (i) holds. Because all the maps θ_{m} are homomorphisms, it is enough to check this in the case where u is an algebra generator, i.e. , E_{i}, F_{i}, K_{i}K_{i+1}^{−1} or K_{i}^{−1}K_{i+1}.

For each such element u, we need to check that

ψ_{n,r+in}(θ_{r+(i+1)n}(u)) = θ_{r+in}(u)

for each natural number i. This is an easy case-by-case check which uses part (3) of Theorem 6.3.5 and the definition of θ. We calculate explicitly the case of K_{i}K_{i+1}^{−1} as an example.

We see from the definition of θ that

θ_{r+(i+1)n}(K_{i}K_{i+1}^{−1}) = ∑_{D∈𝔻_{r+(i+1)n}} v^{d_{i}−d_{i+1}}[D].

Applying part (3) of Theorem 6.3.5, we find that ψ_{n,r+(i+1)n} takes this element to

∑_{D∈𝔻_{r+in}} v^{(d_{i}+1)−(d_{i+1}+1)}[D] = ∑_{D∈𝔻_{r+in}} v^{d_{i}−d_{i+1}}[D] = θ_{r+in}(K_{i}K_{i+1}^{−1}).

This happens because ψ_{n,r+(i+1)n} takes [D] to [D − I] if this makes sense, and to 0 otherwise. The other cases are similar or easier.

Remark This construction fails for U(gl_{n})! Putting the element K_{i} ∈ U(gl_{n}) through the check in the proof of Lemma 6.4.10 does not give the desired result.

The next aim is to prove that θ̂ is injective.

Lemma 6.4.11 We have

⋂_{r=0}^{∞} ker θ_{r} ∩ U(sl_{n}) = 0.

Proof Suppose u ∈ U = U(sl_{n}) lies in ∩ ker θ_{r} for all r. Then, because S_{v}(n, r) is the quotient of U(gl_{n}) which acts faithfully on tensor space V^{⊗r}, u must annihilate V^{⊗r} for all values of r. It is known (see for example §5.1) that any finite-dimensional simple highest weight module L for U(gl_{n}) (and hence for U(sl_{n})) can be embedded in V^{⊗r} for suitable r. Thus u acts as zero on any finite-dimensional simple highest-weight module of U(sl_{n}).

Lusztig [L3, Proposition 6.3.6] proves that an integrable U-module M is a sum of finitedimensional simple highest weight U-modules. Therefore u annihilates M. We now see from [L3, Proposition 3.5.4] that u = 0, which completes the proof.

Theorem 6.4.12 The map θ̂ is a monomorphism of algebras from U(sl_{n}) into Ŝ_{v}(n). The generalised cell modules for Ŝ_{v}(n) are in natural bijection with the simple highest weight modules for U(sl_{n}). Proof The fact that θ̂ is a monomorphism of algebras follows from Lemmas 6.4.10 and 6.4.11.

The link between the two representation theories goes as follows.

Let μ = (μ_{1}, ⋯, μ_{n−1}), where the elements μ_{i} are natural numbers. Define λ_{j} = ∑_{i=j}^{n−1} μ_{i} for each 1 ≤ j < n, and λ_{n} = 0. This means that μ_{i} = λ_{i} − λ_{i+1} and λ ∈ Λ^{∘}. Set r = ∑_{i=1}^{n} λ_{i}.

The element μ is a dominant weight and can be identified with a typical simple highest weight module M for U(sl_{n}). It can be lifted to a simple module M′ for U(gl_{n}) where the action of K_{i} on the highest weight vector z_{λ} (the same notation as in §5) is given by

K_{i}.z_{λ} = v^{λ_{i}}.z_{λ}.

We know from Theorem 5.3.1 and related results that M′ can be identified with the v-Weyl module W_{v}^{λ} for S_{v}(n, r), with r defined as above. Using a version of Proposition 6.2.3 suited to S_{v}(n, r) rather than S_{q}(n, r), we can identify this module with the cell module W (λ). Using Proposition 6.4.7 (ii), one finds that this module corresponds to the generalised cell module of shape λ for Ŝ_{v}(n, r′), where r′ and r are congruent modulo n.

Finally one can identify the module with the corresponding generalised cell module of shape λ for Ŝ_{v}(n).

All the generalised cell modules of Ŝ_{v}(n) turn up in this way.

Definition Let U_{A}(sl_{n}) be the A-subalgebra of U(sl_{n}) generated by all E_{i}^{(c)}, F_{i}^{(c)}, K_{i}K_{i+1}^{−1} and

\left[{K_{i}K_{i+1}^{−1}; 0 \atop c}\right] ,

as 1 ≤ i < n and c ∈ ℕ.

Lemma 6.4.13 The algebra U_{A}(sl_{n}) agrees with the A-form of U(sl_{n}), U_{A}^{−} ⊗ ′U^{0} ⊗ U_{A}^{+}, where ′U^{0} is as given in [D4, §2]. Proof The only nontrivial step is checking that

\left[{K_{i}K_{i+1}^{−1}; 0 \atop c}\right]

lies in ′U^{0}. Since the algebra given is an A-algebra, it must be fixed setwise by the ℤ-algebra homomorphism which sends v to v^{−1}. Applying this map to the given element changes it into

\left[{K_{i}^{−1}K_{i+1}; 0 \atop c}\right] ,

which lies in ′U^{0}.

Corollary 6.4.14 The monomorphism θ̂ restricts to an algebra monomorphism from U_{A}(sl_{n}) to Ŝ_{v}(n). Proof This follows from Lemma 6.4.13, which implies that if u ∈ U_{A}(sl_{n}) then θ_{r}(u) ∈ S_{v}(n, r). (As usual, S_{v}(n, r) refers to our A-form of the v-Schur algebra.)

Finally, we look at the relationship between Ŝ_{v}(n) and the algebra U̇ of type A which was introduced in [L3, §23].

Definition Let Λ̇ be the set of n-tuples of natural numbers (λ_{1}, …, λ_{n}) such that at least one of the λ_{i} is zero.

Lemma 6.4.15 There is a natural bijection between Λ̇ and the set X_{n} of all (n − 1)-tuples μ_{1}, …, μ_{n−1} of integers. This satisfies μ_{i} = λ_{i} − λ_{i+1}.

Proof Clearly each λ determines a μ. Going the other way, we see that each μ clearly determines a λ up to adding an integer multiple of ω. The condition that all the λ_{i} are nonnegative and at least one is zero determines this multiple uniquely.

Definitions To each λ ∈ Λ̇ (corresponding as above to the (n − 1)-tuple of integers μ), we associate an element e_{μ} of Ŝ(n) as follows. Define r′ = ∑_{i=1}^{n} λ_{i}, and let r ≡ r′ (mod n) be such that 0 ≤ r < n. Define k to be the natural number satisfying r′ = r + kn. Let D be the diagonal matrix whose (i, i) entry is λ_{i}. Let (x_{i}) be the element of Ŝ_{v}(n, r) given by setting

x_{i} := \{ \begin{matrix}0 & if i < k, \\ [D + (i − k)I] & otherwise. \end{matrix}

(It follows from part (3) of Theorem 6.3.5 that this is an element of S_{v}(n, r).) We denote by e_{μ} the corresponding element of Ŝ(n).

For each integer i such that 1 ≤ i < n we define, following [L3, §2], the element i′ ∈ X_{n} (where X_{n} is as in Lemma 6.4.15) to be the i-th column of the Cartan matrix corresponding to U(sl_{n}), namely

(i′)_{j} := \{ \begin{matrix}2 & if j = i, \\ −1 & if |j − i| = 1, \\ 0 & otherwise. \end{matrix}

Lemma 6.4.16 We regard U(sl_{n}) as a subalgebra of Ŝ_{v}(n) as in Theorem 6.4.12. The following relations hold in Ŝ_{v}(n):

e_{μ}e_{μ′} = δ_{μ,μ′}e_{μ}  (1)
E_{i}e_{μ} = e_{μ+i′}E_{i}  (2)
F_{i}e_{μ} = e_{μ−i′}F_{i}  (3)
K_{i}K_{i+1}^{−1}e_{μ} = v^{μ_{i}}e_{μ}  (4)
e_{μ}K_{i}K_{i+1}^{−1} = v^{μ_{i}}e_{μ}  (5)

Proof All of these are easy consequences of the definitions and the multiplication in the v-Schur algebra.

Definitions In [L3, §23], Lusztig defines a ℚ(v)-algebra, U̇, and proves that a basis for it is given by the set consisting of elements

b^{−}1_{μ}b^{+},

as b^{−} ranges over a basis for U^{−}, b^{+} ranges over a basis for U^{+} and μ ranges over X_{n}. Lusztig proves that the following relations hold in U̇:

1_{μ}1_{μ′} = δ_{μ,μ′}1_{μ}  (6)
E_{i}1_{μ} = 1_{μ+i′}E_{i}  (7)
F_{i}1_{μ} = 1_{μ−i′}F_{i}  (8)
K_{i}K_{i+1}^{−1}1_{μ} = v^{μ_{i}}1_{μ}  (9)
1_{μ}K_{i}K_{i+1}^{−1} = v^{μ_{i}}1_{μ}  (10)

The multiplication in U̇ is inherited in a natural way (explained in [L3, §23]) from the multiplication in U(sl_{n}). It is not hard to see that the relations (6) to (10) together with the usual relations in U(sl_{n}) are sufficient to express the product of two basis elements as a (finite) linear combination of others. Thus we can determine the structure constants of U̇ with respect to the given basis.

The algebra U̇ has an A-form, U̇_{A}, which has as a free A-basis the set b^{−}1_{μ}b^{+} where b^{−} and b^{+} range over A-bases for U_{A}^{−} and U_{A}^{+} respectively, and μ ranges over X_{n}.

It will be convenient to reparametrise this basis by certain pairs of n × n matrices as follows. We work with the basis B^{−} of U^{−} and B^{+} of U^{+} as in Proposition 1.5. This gives us a basis

of U̇ consisting of elements b = b^{−}1_{μ}b^{+}, where b^{−} ∈ B^{−}, b^{+} ∈ B^{+} and μ ∈ X_{n}.

We define a weakly positive pair of matrices ⟨M, M′⟩ to be matrices with non-negative integer entries satisfying the following conditions:

(i) M is lower triangular; (ii) M′ is upper triangular; (iii) the product [M][M′] is a v-codeterminant for some v-Schur algebra S_{v}(n, r); (iv) at least one of the matrices has a zero on the diagonal.

Lemma 6.4.17 There is a natural bijection between basis elements b^{−}1_{μ}b^{+} of U̇ and weakly positive pairs of matrices. Proof Let ⟨M, M′⟩ be a weakly positive pair of matrices. Let λ be the n-tuple of natural numbers defined by setting λ_{i} to be the sum of the i-th column of M. (This is the same as the sum of the i-th row of M′, by part (iii) of the definition of weakly positive.) Let m be such that λ_{m} ≤ λ_{i} for all i. Then λ − λ_{m}.ω lies in Λ̇. Let μ ∈ X_{n} correspond to λ − λ_{m}.ω as in Lemma 6.4.15. We define b^{−} in B^{−} according to the entries below the diagonal in M, as in Proposition 4.1.7. We define b^{+} according to the entries above the diagonal in M′, as in Proposition 4.1.3. This produces a basis element b^{−}1_{μ}b^{+} of U̇.

We now describe the inverse of this process. Choose a basis element b^{−}1_{μ}b^{+} of U̇. Find the element λ ∈ Λ̇ corresponding to μ as in Lemma 6.4.15. Let X be the lower triangular n × n matrix with integer entries as follows. The entries below the diagonal in X are determined by b^{−} as in Proposition 4.1.7. The diagonal entries may be negative, and are defined so that the sum of column i of X is λ_{i}. We also define X′ to be the upper triangular n×n matrix with integer entries as follows. The entries above the diagonal are determined by b^{+} as in Proposition 4.1.3. The diagonal entries (which may be negative) are chosen in the unique way which makes the sum of row i of X′ equal to λ_{i}. Since λ_{i} ∈ Λ̇, one of the matrices X or X′ must have a nonpositive entry on the diagonal. Let m be the smallest such entry (possibly zero). Now define M = X − m.I and M′ = X′ − m.I. It is easily checked that M is lower triangular, M′ is upper triangular, the column sums of M are the row sums of M′ (so [M][M′] is a v-codeterminant for sum S_{v}(n, r)) and one of M and M′ has a zero entry on the diagonal. This therefore produces a weakly positive pair of matrices in a way which can be checked to be the inverse of the process in the preceding paragraph.

We will now index the basis of U̇ by weakly positive pairs of matrices ⟨M, M′⟩. The basis element corresponding to ⟨M, M′⟩ will be denoted by Ċ(M, M′)

Definition Let ⟨M, M′⟩ be a weakly positive pair of matrices. We define an element Ŷ(M, M′) of Ŝ_{v}(n) as follows. Set r to be the sum of the entries of M (or equivalently of M′). Define r′ such that r′ ≡ r (mod n) and 0 ≤ r′ < n. Let k be such that r = r′ + kn.

The element Ŷ(M, M′) is the element of Ŝ_{v}(n) identified with that element of Ŝ_{v}(n, r′) given by the sequence (x_{i}) where

x_{i} := \{ \begin{matrix}0 & if i < k, \\ [M + (i − k)I][M′ + (i − k)I] & otherwise. \end{matrix}

One checks easily from Theorem 6.3.5 that this is an element of Ŝ_{v}(n). (This works because of the triangular axioms for weakly positive pairs.)

It should be noted that all elements Ĉ(T, T′) turn up naturally in this way. It is not in general true that any such element will have the form Ĉ(T, T′), but if this is the case, we will call the basis element Ċ(M, M′) standard.

Before the proof of the main theorem, we require a certain technical result about classical Schur algebras.

Lemma 6.4.18 Let [L][U] be a codeterminant for some (unquantized) Schur algebra S(n, r), where L is a lower triangular matrix and U is an upper triangular matrix. Let w_{L} be the sum of all the entries below the diagonal of L, and let w_{U} be the sum of all the entries above the diagonal of U. Choose w to be greater than or equal to the maximum of w_{L} and w_{U}.

A unique basis element [A] of S(n, r + wn) exists with the following properties:

1. The entries below the diagonal of A are the same as those of L.

2. The entries above the diagonal of A are the same as those of U.

3. The column sums of A are the same as those of U + w.I.

4. The row sums of A are the same as those of L + w.I.

Consider the product [L+w.I][U+w.I] in the Schur algebra S(n, r+wn). Expanding the product in terms of the natural basis of S(n, r + wn) (not the codeterminant basis), the basis element [A] occurs with nonzero coefficient. All other basis elements occurring correspond to matrices of strictly larger trace. We call the element [A] the leading term of the product [L + w.I][U + w.I]. Proof Uniqueness of the matrix A is clear given existence. If we can show that a basis element [A] occurring in the given product satisfies properties 1 and 2 then we have proved the first assertion, because properties 3 and 4 follow automatically from weight space considerations.

The fact that [A] occurs with nonzero coefficient is a corollary of Schur’s product rule and the choice of the number w, as is the fact that other basis elements occurring have larger trace. This is probably best illustrated with an example.

Let n = r = 3, let L be the matrix

( \begin{matrix}1 & 0 & 0 \\ 2 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

and let U be the matrix

( \begin{matrix}0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

In this case, w_{L} = 2, w_{U} = 3. We choose w = 3. One can check that [L][U] is nonzero by checking the row and column sums of each matrix. We now appeal to the tensor matrix space ideas of §2.1. The element of T^{r+wn}(M_{n}) corresponding to [L+w.I] consists of sums of permutations of the tensor

e_{11} ⊗ e_{21} ⊗ e_{21} \underbrace{⊗e_{11} ⊗ e_{22} ⊗ e_{33}}_{w times}

and the element corresponding to [U + w.I] consists of sums of permutations of

e_{13} ⊗ e_{13} ⊗ e_{13} \underbrace{⊗e_{11} ⊗ e_{22} ⊗ e_{33}}_{w times} .

We have chosen w sufficiently large so that it is possible for two such permutations of tensors to multiply together (and give a nonzero result) so that no non-diagonal matrix unit in one combines with a non-diagonal matrix unit in the other. This gives rise to a basis element [A] with properties 1 and 2. In this case, A is given by

( \begin{matrix}1 & 0 & 3 \\ 2 & 3 & 0 \\ 0 & 0 & 3 \end{matrix} ) .

The other basis elements arise from two non-diagonal matrix units multiplying with each other. This gives rise to strictly more diagonal matrix units in the result, and a basis element of larger trace.

Definition We define the A-module ˆ U̇ to be the set of all formal infinite sums

∑ c_{M,M′} Ċ(M, M′),

where the pairs ⟨M, M′⟩ are weakly positive and the element c_{M,M′} ∈ A is the coefficient of Ċ(M, M′) in the sum.

Theorem 6.4.19 Regard U(sl_{n}) as a subalgebra of Ŝ_{v}(n) as in Theorem 6.4.12.

(a) The A-linear map θ̇ from U̇_{A} to Ŝ_{v}(n) which takes Ċ(M, M′) to Ŷ(M, M′) is a homomorphism of A-algebras whose image is dense with respect to the topology on Ŝ_{v}(n). Furthermore, θ̇ is injective.

(b) The A-module map θ̇ extends in a natural way to ˆ U̇. The kernel of θ̇ applied to ˆ U̇ consists of infinite A-combinations of elements of the form

Ċ(M, M′) − ∑_{⟨A,A′⟩} c_{A,A′} Ċ(A, A′)

where Ċ(M, M′) is weakly positive but not standard, the elements Ċ(A, A′) are standard, and the coefficients c_{A,A′} lie in A.

Proof

We first prove part (a). We find from [L3, Lemma 23.2.2] and the relation (7) from earlier that U̇ is generated as an algebra by the elements 1_{μ}E_{i}^{(c)} and F_{i}^{(c)}1_{μ}, for all nonnegative integers c and elements μ ∈ X_{n}.

Each element 1_{μ} in U̇ is equal to an element Ċ(D, D), where D is diagonal. It therefore maps to the idempotent Ŷ(D, D), which is equal to e_{μ}.

One checks that the element

1_{μ}E_{i}^{(c)} = Ċ(D, D′ + cE_{i,i+1})

maps under θ̇ to

e_{μ}E_{i}^{(c)} = Ŷ(D, D′ + cE_{i,i+1}).

Here, D is such that

θ̇(1_{μ}) = Ċ(D, D),

and as usual, E_{i,i+1} is the matrix unit with 1 in the (i, i+1)-place and zeros elsewhere. The diagonal matrix D′ is determined by the property that the sum of row h of D′ + cE_{i,i+1} is equal to D_{hh}. Analogous remarks hold for the element F_{i}^{(c)}1_{μ}.

We now have to check all the relations hold in the image. The usual relations of the quantized enveloping algebra certainly hold because of Theorem 6.4.12. The new relations (6) to (10) in U̇ hold because of Lemma 6.4.16 and the similarity between the relations (1) to (5) in that lemma with the relations (6) to (10).

The “density” part of the argument holds because elements of Ŝ_{v}(n) can be approximated arbitrarily “closely” (in the topology arising from the sets B_{k}) by finite A-linear combinations of basis elements Ŷ(M, M′).

We now need to prove that θ̇ is injective. Suppose to the contrary that there exists an element of U̇ given by

u = ∑_{⟨M,M′⟩} c_{M,M′} Ċ(M, M′)

(where the scalars c_{M,M′} are not all zero) which maps under θ̇ to zero. (Note that the sum is finite.) We may assume that this sum is as short as possible. Without loss of generality, we may assume all the Ċ(M, M′) are mapped by θ̇ into the same Ŝ_{v}(n, r) for a fixed r. Also without loss of generality we may assume all the scalars c_{M,M′} are elements of ℤ[v], and furthermore we may assume that they are not all divisible by v − 1. Thus by replacing v by 1, we obtain a counterexample to the corresponding unquantized problem. We may therefore concentrate on proving this.

With the aid of Lemma 6.4.18, we find that by taking a sufficiently large value of k, the k-th component of θ̇(Ċ(M, M′)) (i.e. that corresponding to S(n, r + kn)) contains (with nonzero coefficient) the leading term of the k-th component of the product corresponding to Ŷ(M, M′). We can choose k so that this happens for every Ċ(M, M′) appearing in the sum u. (We have used the fact that the sum u is finite here. This is important.) Now consider the basis elements Ċ(M, M′) appearing which give rise to leading terms of minimal trace. By “minimal”, we mean minimal among the set of traces of leading terms arising from elements Ċ(M, M′) appearing in u. One sees easily from the definition of leading term that different weakly positive pairs give rise to different leading terms. It is therefore not possible for any cancellation to occur among these leading terms of minimal trace, because Lemma 6.4.18 shows that other basis elements occurring have larger traces. We are forced to conclude that u is not in fact mapped to zero by θ̇, which proves the claim that θ̇ is injective.

We now tackle part (b). It is possible to extend θ̇ naturally to the infinite sums shown because for any open set B_{k}, all but finitely many basis elements Ċ(M, M′) ∈ U̇ map into B_{k} under θ̇. This means that the finite partial sums of an element in ˆ U̇ will map under θ̇ to a Cauchy sequence in Ŝ_{v}(n). Because of the completeness of Ŝ_{v}(n), we can now extend θ̇ to ˆ U̇.

We know θ̇ : ˆ U̇ → Ŝ_{v}(n) is surjective because each standard v-codeterminant of Ŝ_{v}(n) is the image of a standard basis element Ċ(M, M′).

Observe that the standard basis elements of U̇ ⊂ ˆ U̇ map to elements Ĉ(T, T′) of Ŝ_{v}(n). The non-standard basis elements map to elements which are images of elements u ∈ ˆ U̇ expressible as infinite A-combinations of standard elements Ċ(A, A′). Thus for any weakly positive non-standard pair ⟨M, M′⟩, one can find an infinite sum as (in the statement) lying in ker θ̇. Note that Ċ(M, M′) and all the Ċ(A, A′) appearing map under θ̇ to elements in Ŝ_{v}(n, r) for some fixed r.

Finally we note that infinite A-combinations of elements as given in the statement are actually elements of ˆ U̇. This happens because each standard basis element of U̇, Ċ(A, A′), can only appear in finitely many infinite sums corresponding to different weakly positive pairs ⟨M, M′⟩ as in the statement of part (b). This follows from the fact that only finitely many elements Ċ(M, M′) lie outside any given open set B_{k}, and the observation that if θ̇(Ċ(M, M′)) lies in the set B_{k}, then so do all the elements θ̇(Ċ(A, A′)), where Ċ(A, A′) appears in the sum. (If the latter were not the case, one could use Proposition 6.4.6 (ii) to contradict the basis theorem for standard v-codeterminants.) The total coefficient of a standard element Ċ(A, A′) in any such infinite combination of infinite sums indexed by weakly positive pairs will therefore be a well-defined element of A. (The analogous observation for non-standard elements Ċ(M, M′) is trivial because each such appears in only one of the infinite sums in the statement.)

Concluding remarks

In [L3, §29], Lusztig essentially proves that U̇ is cellular, and that its representation theory is the highest weight portion of the representation theory of U(sl_{n}). This means that U̇ and Ŝ_{v}(n) have the same absolutely irreducible finite-dimensional modules.

By combining the results of Theorem 6.4.19 and Proposition 6.4.7 (ii) and using the direct sum decomposition of Ŝ_{v}(n), one finds that the v-Schur algebras S_{v}(n, r) are canonically quotients of the algebra U̇. We can therefore reformulate the main results of §4 with U̇ in the role of U(gl_{n}). This seems to be a more natural situation than the one involving U(gl_{n}): for example, one does not have the slightly messy problem of calculating U^{0} ∩ ker θ which was dealt with in Proposition 4.1.12.

An application of the fact that S_{v}(n, r) is a quotient of ˆ U̇ (as an A-module) is that all the coefficients c_{A,A′} in the statement of Theorem 6.4.19 (b) are calculable, because they are the same as certain of the coefficients arising in the straightening formula for v-Schur algebras described in §3. To see this, take one of the expressions for an element of ker θ̇ as described in Theorem 6.4.19 (b), and pass to the quotient S_{v}(n, r). The nonstandard element of U̇ occurring maps to a distinguished, but not standard, v-codeterminant (in the sense of §4) or to zero; the standard elements map to standard v-codeterminants, or to zero. For each nonzero element c_{A,A′}, one can choose a suitable r such that Ċ(A, A′) will not map to zero under the quotient map, but instead to c_{A,A′} times a certain v-codeterminant. This means that our description of the elements of the kernel of the canonical map from ˆ U̇ to Ŝ_{v}(n) should be regarded as the limiting case of the straightening formula for distinguished v-codeterminants as described in Corollary 4.2.6.

One can also find results for U̇ analogous to the main results in §5 dealing with the representation theory of U(gl_{n}). Again, the proofs are more elegant.

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