Title: Positivity Properties for q-Schur Algebras Author: Richard M. Green Institution: Mathematical Institute, Oxford University Correspondence to: Dr R.M. Green, Mathematical Institute, 24–29 St. Giles’, Oxford OX1 3LB, England Email address: greenr@maths.ox.ac.uk

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POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS

R.M. Green Mathematical Institute Oxford University 24–29 St. Giles’ Oxford OX1 3LB England E-mail: greenr@maths.ox.ac.uk

Abstract. We prove Du’s positivity conjecture for the canonical basis of the q-Schur algebra, using elementary arguments and the positivity result for Lusztig’s canonical basis for U_{q}^{+}(sl_{n}). We also describe a family of subalgebras of the q-Schur algebra, each of which is spanned by the canonical basis elements it contains.

0. Introduction

The q-Schur algebra, S_{q}(n, r) was first introduced by Dipper and James in [2]. Du [4] introduced a canonical basis for this algebra, which is analogous to the bases of Kazhdan and Lusztig for Hecke algebras. Du conjectures [7, §5] that the structure constants with respect to this canonical basis are Laurent polynomials in the indeterminate v := q^{\frac{1}{2}} with coefficients in ℕ. Du remarks that such a result ought to have a proof involving intersection cohomology, but that there should also be some algebraic or combinatorial approach to the problem. The main purpose of this paper is to prove Du’s conjecture. The methods used are elementary in that they do not explicitly involve methods from intersection cohomology, but we shall require a deep result due to Lusztig, namely the positivity result for Lusztig’s canonical basis of the quantized enveloping algebra of type A. Furthermore, the positivity result for q-Schur algebras is equivalent to Lusztig’s result in the sense that an elementary proof of one would give an elementary proof of the other.

The rest of the paper falls into four parts, the first of which is expository.

Research supported by the E.P.S.R.C.

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 3

In §2, we study a certain class of subalgebras of S_{q}(n, r), each of which turns out to be spanned by the canonical basis elements it contains.

In §3 we analyse a certain Lusztig–Dyer basis associated with the quantized enveloping algebra of type A. This turns out to be connected to the natural basis of the Borel subalgebra S_{q}^{+}(n, r) of S_{q}(n, r).

In §4, we prove the positivity result, first proving it for the Borel subalgebra, and then extending it to the whole algebra.

1. Du’s Canonical Basis

In this paper, we shall denote the ring of Laurent polynomials ℤ[v, v^{−1}] by A. We shall also be working with a certain A-form of the q-Schur algebra, also denoted by S_{q}(n, r), as no confusion is likely to arise. As always, the indeterminates q and v are related via q = v^{2}.

The canonical basis is a free A-basis for S_{q}(n, r). It first appeared in [4], where it was defined in the following way.

Let W = S_{r} be the symmetric group on r letters, and let S be the set of basic transpositions

(1, 2), (2, 3), ⋯, (r − 1, r).

Let λ be a composition of r, namely

λ = (λ_{1}, ⋯, λ_{m})

with λ_{1} + ⋯ + λ_{m} = r and λ_{i} ≥ 0 for all i. Thus λ determines naturally a subset I(λ) of S. We will write W_{λ} for the parabolic subgroup of W associated with this subset, and D_{λ,μ} for the (unique) set of double coset representatives of minimal length corresponding to the double cosets W_{λ}\W/W_{μ}. We also write D_{λ,μ}^{+} for the set of double coset representatives of maximal length.

We denote by H the Hecke algebra over A corresponding to W. This is a free A-module with basis {T_{w}}_{w∈W} satisfying the relations

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) = ℓ(w) + 1, \\ qT_{sw} + (q − 1)T_{w} & otherwise, \end{matrix}

where s ∈ S and ℓ(w) denotes the length of w. If D is a subset of W, we write

T_{D} := ∑_{x∈D} T_{x}.

We write x_{λ} for T_{W_{λ}}. Following [3], the q-Schur algebra is defined as

S_{q}(n, r) := End_{H} ( ⨁_{λ∈Λ(n,r)} x_{λ}H) .

(We are regarding x_{λ}H as a right module for H via right multiplication.) When q is replaced by 1, we recover the classical Schur algebra S(n, r) which was described in [9].

For λ, μ ∈ Λ(n, r) and w ∈ D_{λ,μ}, we define the element ϕ_{λ,μ}^{w} ∈ S_{q}(n, r) by

ϕ_{λ,μ}^{w}(x_{ν}h) = δ_{μ,ν} ∑_{x∈W_{λ}wW_{μ}} T_{x}h,

where δ_{μ,ν} is the Kronecker delta. It was proved in [3] that the set

{ϕ_{λ,μ}^{w} : λ, μ ∈ Λ(n, r), w ∈ D_{λ,μ}}

forms a basis of S_{q}(n, r). We will call this the natural basis of S_{q}(n, r).

We can define the canonical basis from this basis (using the same indexing set) by the formula (given in [4, (2.c)])

θ_{λμ}^{w} = v^{ℓ(w_{0,μ})} ∑_{z∈D_{λ,μ}} α_{z,w}ϕ_{λ,μ}^{z},

where w_{0,μ} is the longest element of W_{μ} and α_{z,w} is given by

v^{−ℓ(w^{+})}P_{z^{+},w^{+}}.

Here the polynomial P is the usual Kazhdan–Lusztig polynomial, and we have denoted by w^{+} the longest element in the same W_{λ}–W_{μ} double coset by w.

We also introduce the set M(n, r) of n × n matrices with nonnegative integer coefficients whose sum is r. With each triple (λ, w, μ) where λ, μ ∈ Λ(n, r)) and w ∈ W, one can associate an element of M(n, r) in such a way that the triples indexing the natural basis are in bijection with the set M(n, r). It is important

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 5

for the purposes of §2 to understand this correspondence. Let I(n, r) be the set of r-tuples with integer entries from 1 to n inclusive. If ν ∈ Λ(n, r), we write ℓ(ν) for the element of I(n, r) given by

(i_{1}, i_{2}, …, i_{r}),

where i_{1} ≤ ⋯ ≤ i_{r} and s occurs ν_{s} times. Given a triple (λ, w, μ), we define the corresponding matrix A ∈ M(n, r) by stipulating that A_{i,j} is the multiplicity of the ordered pair ⟨i, j⟩ in the family

{⟨ℓ(λ)_{w(i)}, ℓ(μ)_{i}⟩},

as i ranges from 1 up to r. We will sometimes write θ_{A} for θ_{λ,μ}^{w} in this case.

2. Good Subalgebras

Following [10, §2.3], we now define “good” subalgebras of the q-Schur algebra S_{q}(n, r). Consider a set G of matrix units of M_{n} with the following two properties:

1. If e_{ij} ∈ G and e_{jk} ∈ G then e_{ik} ∈ G;

2. If a > c and b < d, and if e_{ab} ∈ G and e_{cd} ∈ G, then also e_{ad} ∈ G and e_{cb} ∈ G. Now consider the subset G(n, r) of M(n, r) defined by the condition

A ∈ G(n, r) iff (A_{ij} ≠ 0 ⇒ e_{ij} ∈ G).

Definition Let G be a good set of matrix units. We denote by S_{q}(G) the subspace of S_{q}(n, r) spanned by the set {θ_{A} : A ∈ G(n, r)}.

Lemma 2.1 Let G be a good set of matrix units, and let A ∈ G(n, r). Write θ_{λ,μ}^{w} for θ_{A} (where w is any element of W), and let w′ ≤ w in the (strong) Bruhat order. Let B be such that θ_{B} = θ_{λ,μ}^{w′}. Then B ∈ G(n, r). Proof It was mentioned in [8, Lemma 2.2] that w_{B} ≤ w_{A} if and only if the sequence (w_{A}(1), …, w_{A}(r)) is obtainable from the sequence (w_{B}(1), …, w_{B}(r)) by a sequence of transpositions (i, j) where i < j and i occurs to the left of j in (w_{B}(1), …, w_{B}(r)).

6 R.M. GREEN

Now consider an arbitrary ascending sequence such as ℓ(λ) on which w_{B} and w_{A} act. (The case of the sequence (1, 2, …, r) was done in the last paragraph.) It follows by using the above techniques and omitting irrelevant transpositions that if w_{B} ≤ w_{A} then the sequence

(w_{A}(ℓ(λ)_{1}), …, w_{A}(ℓ(λ)_{r}))

is obtainable from the sequence

(w_{B}(ℓ(λ)_{1}), …, w_{B}(ℓ(λ)_{r}))

by a sequence of transpositions (i, j) where i < j and i occurs to the left of j in (w_{B}(ℓ(λ)_{1}), …, w_{B}(ℓ(λ)_{r})).

Using the correspondence between the two forms of the basis, it is now not hard to see that the matrix B can be obtained from the matrix A in a sequence

(A = C_{0}, C_{1}, C_{2}, …, C_{m}, C_{m+1} = B)

where each of the C_{i} lies in M(n, r) and for each i, C_{i+1} − C_{i} is a matrix of the form

e_{ab} + e_{cd} − e_{ad} − e_{cb},

where a < c and b < d. Axiom 2 for a good set of matrix units now shows that all the matrices C_{i} lie in G(n, r) (because e_{ab} and e_{cd} must be elements of G if e_{ad} and e_{cb} are); in particular, B does. □

Theorem 2.2 The A-subspace S_{q}(G) of S_{q}(n, r) is a subalgebra of S_{q}(n, r) of dimension

\binom{m + r − 1}{r},

where m is the cardinality of G. Proof This is a reworking of [10, Theorem 2.3.3], which says the same thing but which starts with a different definition of S_{q}(G). It therefore suffices to prove that the two definitions are equivalent.

The definition of S_{q}(G) in [10, §2.3] is the A-module spanned by the natural basis elements {[A] : A ∈ G(n, r)} for the good set G. (The element [A] is a certain

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 7

power of v times the element e_{A}.) Clearly this has the same dimension as the S_{q}(G) we have just defined, so it is enough to show that if A ∈ G(n, r) then θ_{A} is expressible as a linear combination of the set {[A′] : A′ ∈ G(n, r)}. To prove this, we use Lemma 2.1 and the definition of the canonical basis.

By definition, we know that

θ_{λμ}^{w^{+}} = v^{ℓ(w_{0,μ})} ∑_{z∈D_{λ,μ}} α_{z,w}ϕ_{λ,μ}^{z^{+}},

where w_{0,μ} and α_{z,w} are as in §1. Because P_{z^{+},w^{+}} (and hence α_{z,w}) is only nonzero if z^{+} ≤ w^{+}, we can apply Lemma 2.1 to this expression and deduce that if the matrix indexing θ_{λ,μ}^{w^{+}} is in G(n, r), then so are all the matrices indexing the terms on the other side with respect to the natural basis. This proves the desired result. □

Remarks

One can now deduce from the results about Zariski closures in the appendix of [4] that good subalgebras have a geometric interpretation in the approach of Beilinson, Lusztig and MacPherson [1]. They correspond to certain closed subalgebras of the q-Schur algebra with respect to the Zariski topology (i.e. subalgebras which which are unions of closed orbits, since there are only finitely many orbits) provided that r > 1. This follows because the ordering ≤ defined in [1] in terms of the Zariski closure is essentially the same as the strong Bruhat order.

It is also obvious that a positivity property for the canonical basis of any good subalgebra would follow from the positivity property of S_{q}(n, r).

In §3 we will require the following result. Corollary 2.3 The Borel subalgebra S_{q}^{+}(n, r) spanned by all upper triangular elements [U] is spanned by the canonical basis elements it contains, that is, the cones corresponding to upper triangular matrices. Proof Note that the set {e_{ij} : i ≤ j} is good. For any θ_{A}, it will be true that θ_{A} lies in S_{q}^{+}(n, r) if and only if all the terms in its expression in terms of the natural basis do. We now see from Theorem 2.2 that this will be true if and only if the matrix A is upper triangular. The assertion is now clear from dimension considerations. □

8 R.M. GREEN

3. Lusztig–Dyer Bases

The aim of §3 is to understand explicitly the connection between a certain Lusztig–Dyer basis of the quantized enveloping algebra U^{+}(sl_{n}) and the natural basis of the q-Schur algebra S_{q}(n, r). Once this is understood, we can compare Lusztig’s canonical basis for U^{+}(sl_{n}) with the canonical basis for S_{q}(n, r). It will turn out that there is a close connection.

We then use this connection to deduce positivity properties for the Borel subalgebra S_{q}^{+}(n, r).

The A-subalgebra U^{+}(sl_{n}) of the quantized enveloping algebra U(sl_{n}) is generated by elements E_{1}, …, E_{n} subject to relations

E_{i}E_{j} = E_{j}E_{i} if |i − j| > 1

and

E_{i}^{2}E_{j} − (v + v^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 if |i − j| = 1.

The first aim (which is the hardest part of this section) is to show that a certain Lusztig–Dyer basis for U^{+}(sl_{n}) agrees with the basis of U^{+}(sl_{n}) which was studied in [10].

We denote by W the symmetric group S_{r} regarded as a Coxeter group of type A_{r−1}. The generators are the simple transpositions s_{p} := (p, p + 1).

We denote by α_{i,j+1} (where i ≤ j) the positive root α_{i} + α_{i+1} + ⋯ + α_{j}.

Let s_{i_{1}}s_{i_{2}} ⋯ s_{i_{N}} be a reduced expression for the longest word w_{0} of W. Associated to this reduced expression is an ordering on the positive roots: the m-th root in the list is

s_{i_{1}} ⋯ s_{i_{m−1}}(α_{i_{m}}),

where the action is the usual action of the Weyl group on the root system and α_{i_{m}} is the i_{m}-th fundamental root.

Lemma 3.1 The word

(s_{n})(s_{n−1}s_{n})(s_{n−2}s_{n−1}s_{n}) ⋯ (s_{1} ⋯ s_{n})

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 9

is a reduced expression for the longest element w_{0} of W. It corresponds to the ordering on the positive roots of sl_{n} given as follows: the root α_{i,j} appears to the left of the root α_{k,l} if and only if i > k or (i = k and j > l). Proof It is well-known (see for example [15]) that the given expression is a reduced expression for the longest element of W. The other assertion can be seen by a straightforward induction on n, using the fact that if α, β, α+β are all positive roots, then the root α + β appears between the other two in the order. (This was mentioned in [11].) □

3.2 The Lusztig–Dyer basis L corresponding to this reduced expression of the longest word was given in [14, 42.1.4] and consists of certain elements

E_{α_{1}}^{(c_{1})}E_{α_{2}}^{(c_{2})} ⋯ E_{α_{N}}^{(c_{N})},

where N is the number of roots in the order α_{1}, …, α_{N}. The notation E_{α}^{(c)} means

\frac{E_{α}^{c}}{[c]!} ,

where [c] := [1][2] ⋯ [c] and

[c] := \frac{v^{c} − v^{−c}}{v − v^{−1}} .

The basis studied in [10] for U^{+}(sl_{n}) had a similar form, except that the corresponding elements Ê_{α_{i,j+1}} were defined recursively via Ê_{α_{i,i+1}} := E_{i} and

Ê_{α_{i,j+1}} := Ê_{α_{i,j}} Ê_{α_{j,j+1}} − v^{−1}Ê_{α_{j,j+1}} Ê_{α_{i,j}}

for |j − i| > 1.

To prove the two bases agree, it is enough to prove that E_{α_{i,j}} = Ê_{α_{i,j}} for all roots α_{i,j}, because the orderings on the “root vectors” are the same in both cases. First, we will prove that they agree up to sign, and then we will prove that they agree. Since we are interested in positivity properties, it is vital that the signs work out correctly.

Lemma 3.3 For each positive root α_{ij},

E_{α_{i,j}} = ±Ê_{α_{i,j}}.

10 R.M. GREEN

Proof In [12, Lemma 2.5], Lusztig proves some identities about certain elements E′_{α_{i,j}} (which are denoted by E_{α_{i,j}}, but we shall not do this, since they aren’t the same as our elements E_{α_{i,j}}). One can easily check that they satisfy

E′_{α_{i,j+1}} = −E′_{α_{i+1,j+1}}E′_{α_{i,i+1}} + v^{−1}E′_{α_{i,i+1}}E′_{α_{i+1,j+1}}.

Lusztig shows [12, Corollary 4.3] that this element E′_{α_{i,j+1}} is equal to T_{w_{α}}(E_{i_{α}}), where T_{w_{α}} is a certain element of the braid group acting on U(sl_{n}). Because E′_{α_{i,j+1}} lies in U^{+}, we invoke [12, 1.3(e)] and deduce that

T_{w_{α}^{−1}}^{−1}(E_{i_{α}}) = (T_{w_{α}}(E_{i_{α}}))^{op},

where op is the opposition anti-automorphism of U.

If we denote the left hand side of this equality by E′′_{α_{i,j+1}}, then the above observation shows that

E′′_{α_{i,j+1}} = −E′′_{α_{i,i+1}}E′′_{α_{i+1,j+1}} + v^{−1}E′′_{α_{i+1,j+1}}E′′_{α_{i,i+1}}.

A routine induction on j − i (using the fact that E_{a} and E_{b} commute when |a − b| > 1) shows that the elements Ê satisfy the following identity:

Ê_{α_{i,j+1}} = Ê_{α_{i,i+1}} Ê_{α_{i+1,j+1}} − v^{−1}Ê_{α_{i+1,j+1}} Ê_{α_{i,i+1}}.

Since Ê_{i,i+1} = E′′_{α_{i,i+1}} = E_{i}, we see by another easy induction on |j − i| (using the above two equations) that

E′′_{α_{i,j+1}} = ±Ê_{α_{i,j}}.

We also find from the definition of the elements E_{α_{i,j+1}} appearing in the Lusztig– Dyer basis that

E_{α_{i,j+1}} := T̃_{w_{α}}(E_{i_{α}}) = ±T_{w_{α}^{−1}}^{−1}(E_{i_{α}}) = ±E′′_{α_{i,j+1}}.

The proof follows from these two observations. □

To prove that our two bases are the same, it is now enough to work with the unquantized situation, as we are only interested in verifying that the signs are correct. This has the advantage of making the calculations much easier.

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 11

The braid group of type A_{n} is generated by elements T̃_{1}, …, T̃_{n−1} subject to the braid relations, namely

T̃_{i}T̃_{j} = T̃_{j}T̃_{i}

when |i − j| > 1 and

T̃_{i}T̃_{j}T̃_{i} = T̃_{j}T̃_{i}T̃_{j}

when |i−j| = 1. Lusztig [14, 37.1.3] defines an action of the braid group on U(sl_{n}). In the case where v = 1 (i.e. the classical case), this proves the identities

T̃_{i}(E_{j}) = −E_{i}E_{j} + E_{j}E_{i}

if |i − j| = 1 and

T̃_{i}(E_{j}) = E_{j}

if |i − j| > 1.

Lemma 3.4 The following identities hold for v = 1:

T̃_{j+1}(Ê_{α_{i,j+1}}) = Ê_{α_{i,j+2}},  (1)
T̃_{i−1}(Ê_{α_{i,j+1}}) = −Ê_{α_{i−1,j+1}}.  (2)

Proof To prove (1) and (2), we induct on m = j − i. The case m = 0 is done by the observations preceding the Lemma. Assume now that m > 0.

We first prove (1). Recall that

Ê_{α_{i,j+1}} = Ê_{α_{i,i+1}} Ê_{α_{i+1,j+1}} − Ê_{α_{i+1,j+1}} Ê_{α_{i,i+1}}.

Lusztig shows that the action of an element of the braid group is an algebra homomorphism. Since m > 0, we see that T̃_{j+1}(Ê_{α_{i,i+1}}) = Ê_{α_{i,i+1}}. Using the inductive hypothesis, we find that

T̃_{j+1}(Ê_{α_{i,j+1}}) = Ê_{α_{i,i+1}} Ê_{α_{i+1,j+2}} − Ê_{α_{i+1,j+2}} Ê_{α_{i,i+1}}.

The identity (1) now follows.

The proof of (2) is similar but uses the identity

Ê_{α_{i,j+1}} = Ê_{α_{i,j}} Ê_{α_{j,j+1}} − Ê_{α_{j,j+1}} Ê_{α_{i,j}}. □

Lemma 3.5 Let v = 1. Then

E_{α_{1,n}} = Ê_{α_{1,n}}.

Proof Let l = n − 1. The definition of E_{α_{1,n}} is

(T̃_{l})(T̃_{l−1}T̃_{l})(T̃_{l−2}T̃_{l−1}T̃_{l}) ⋯ (T̃_{2} ⋯ T̃_{l})(E_{1}).

By repeatedly discarding irrelevant T̃s on the right and commuting other T̃s as far to the right as possible, one sees that this is equal to

T̃_{l}T̃_{l−1} ⋯ T̃_{2}(E_{1}).

Repeated application of Lemma 3.4 (1) now shows that this is equal to Ê_{α_{1,n}}, as required. □

Lemma 3.6 Let v = 1. Then

E_{α_{1}} = Ê_{α_{1}}.

Proof The right hand side is equal to E_{1} by definition. Letting l = n − 1, the left hand side is defined as

(T̃_{l})(T̃_{l−1}T̃_{l}) ⋯ (T̃_{2} ⋯ T̃_{l})(T̃_{1} ⋯ T̃_{l−1})(E_{l}).

Because the last root in the list is known to be α_{l}, and therefore

(s_{l})(s_{l−1}s_{l}) ⋯ (s_{2} ⋯ s_{l})(s_{1} ⋯ s_{l−1})(α_{l}) = α_{1},

the result follows from [13, 1.3(c)]. □

Lemma 3.7 Let v = 1, and suppose 1 < a < n for a ∈ ℕ. Then

(T̃_{2}T̃_{3} ⋯ T̃_{n−1})(T̃_{1}T̃_{2} ⋯ T̃_{a−1})(E_{a}) = −T̃_{1}T̃_{2} ⋯ T̃_{a}(E_{a+1}).

Proof We first claim that if 2 ≤ m ≤ a then

T′ = (T̃_{2}T̃_{3} ⋯ T̃_{m})(T̃_{1}T̃_{2} ⋯ T̃_{a})(E_{a+1}) = (T̃_{2}T̃_{3} ⋯ T̃_{m−1})(T̃_{1}T̃_{2} ⋯ T̃_{a})(E_{a+1}).

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 13

By commuting the T̃_{m} to the right as far as possible, we find that

T′ = (T̃_{2}T̃_{3} ⋯ T̃_{m−1})(T̃_{1}T̃_{2} ⋯ T̃_{m}T̃_{m−1}T̃_{m}T̃_{m+1} ⋯ T̃_{a})(E_{a+1})

Now rewrite T̃_{m}T̃_{m−1}T̃_{m} = T̃_{m−1}T̃_{m}T̃_{m−1} to get

T′ = (T̃_{2}T̃_{3} ⋯ T̃_{m−1})(T̃_{1}T̃_{2} ⋯ T̃_{m−1}T̃_{m}T̃_{m−1}T̃_{m+1} ⋯ T̃_{a})(E_{a+1}).

We now commute the T̃_{m−1} all the way to the right, where it acts trivially on E_{a+1}. It can therefore be discarded. Thus

T′ = (T̃_{2}T̃_{3} ⋯ T̃_{m−1})(T̃_{1}T̃_{2} ⋯ T̃_{m−1}T̃_{m}T̃_{m+1} ⋯ T̃_{a})(E_{a+1}).

This establishes the claim.

Now let

T = (T̃_{2}T̃_{3} ⋯ T̃_{n−1})(T̃_{1}T̃_{2} ⋯ T̃_{a−1})(E_{a}).

By commuting all T̃_{i} where i > a + 1 to the right, we note that

T = (T̃_{2}T̃_{3} ⋯ T̃_{a+1})(T̃_{1}T̃_{2} ⋯ T̃_{a−1})(E_{a}).

Then we commute the T̃_{a+1} to the right to give

T = (T̃_{2}T̃_{3} ⋯ T̃_{a})(T̃_{1}T̃_{2} ⋯ T̃_{a−1})T̃_{a+1}(E_{a}).

Now rewrite T̃_{a+1}(E_{a}) as −T̃_{a}(E_{a+1}). Now we have

−T = (T̃_{2}T̃_{3} ⋯ T̃_{a})(T̃_{1}T̃_{2} ⋯ T̃_{a})(E_{a+1}).

Repeated applications of the initial claim now yield the desired result. □

Definitions

Let Ψ^{+} be the algebra homomorphism from the subalgebra of U(sl_{n}) generated by

{E_{i}, F_{i}, K_{i}, K_{i}^{−1} : i < n − 1}

to the subalgebra of U generated by

{E_{i}, F_{i}, K_{i}, K_{i}^{−1} : i > 1}

14 R.M. GREEN

which takes E_{i} to E_{i+1}, F_{i} to F_{i+1}, K_{i} to K_{i+1} and K_{i}^{−1} to K_{i+1}^{−1}.

Denote by Ψ^{−} be the inverse to this map. It is clear from the nature of the relations of U(sl_{n}) that Ψ^{+} and Ψ^{−} are algebra homomorphisms.

Lemma 3.8 Suppose i > 1 and j < n. Then Ψ^{−}(Ê_{i,j}) = Ê_{i−1,j−1} and Ψ^{+}(Ê_{i,j}) = Ê_{i+1,j+1}. Proof The proof in both cases is a straightforward induction on |j − i|. □

Proposition 3.9 For all positive roots α_{i,j} with 1 ≤ i < j ≤ n,

E_{α_{i,j}} = Ê_{α_{i,j}}.

Proof By the remarks following Lemma 3.3, we need only verify the statement of the Proposition for the case v = 1. The proof is by induction on n. The base case of n = 2 is obvious.

Now consider all the roots except the last (n − 1) in the list. We see that α_{i,j} is such a root if and only if i > 1. The definition of the corresponding E_{α_{i,j}} is of the form

T̃_{i_{1}} ⋯ T̃_{i_{m−1}}(E_{i_{m}}),

where all the entries i_{s} appearing are strictly greater than 1. (Note that in our reduced expression for w_{0}, there is only one occurrence of s_{1}.)

Now consider

E′ = T̃_{i_{1}−1} ⋯ T̃_{i_{m−1}−1}(E_{i_{m}−1}).

The nature of Lusztig’s action of the braid group shows that this is in fact equal to Ψ^{−}(E_{α_{i,j}}). However, we see by reference to the case of (n − 1) that E′ = E_{α_{i−1,j−1}}. Hence

E_{α_{i,j}} = Ψ^{+}(E′) = Ψ^{+}(E_{α_{i−1,j−1}}).

We now use the inductive hypothesis (for the case (n−1)) and Lemma 3.8 to deduce that

Ψ^{+}(E_{α_{i−1,j−1}}) = Ψ^{+}(Ê_{α_{i−1,j−1}}) = Ê_{α_{i,j}}

as required.

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 15

We now turn our attention to the last (n − 1) roots in the list. The longest root is dealt with by Lemma 3.5. The last root is dealt with by Lemma 3.6. The other root vectors are of the form E_{1,a} for 2 < a < n. Denote by E_{i,j,n−1} the Lusztig–Dyer basis element corresponding to α_{i,j} in the case n − 1 (rather than n).

Working from the definition of E_{1,a}, applying Lemma 3.7 and commuting the T̃_{1} appearing to the left, we find that

E_{1,a} = −T̃_{1}(Ψ^{+}(E_{1,a−1,n−1})).

Now by using the inductive hypothesis on n, Lemma 3.8 and Lemma 3.4 (2), we find that

E_{1.a} = −T̃_{1}(Ψ^{+}(Ê_{1,a−1})) = −T̃_{1}(Ê_{2,a}) = Ê_{1,a}.

This completes the proof. □

The reason for going to so much trouble in checking that the two bases of U^{+} agree is that the Lusztig–Dyer basis that we have been studying is closely related to the natural basis {[A] : A ∈ M(n, r)} of S_{q}(n, r). In [6], a certain algebra homomorphism π_{r} : U(sl_{n}) → S_{q}(n, r) was introduced. We use this to relate the Lusztig–Dyer basis to a basis of S_{q}^{+}(n, r), the Borel subalgebra of S_{q}(n, r) spanned by all upper triangular basis elements.

Proposition 3.10 Let X be the set of orthogonal idempotents {[D]} of S_{q}(n, r) parametrised by diagonal matrices in M(n, r). Denote by B the Lusztig–Dyer basis of U(sl_{n}), as defined in §3.2. Then the nonzero elements of the set

{[D].π_{r}(b) : [D] ∈ X, b ∈ B}

are precisely the elements

{[U] ∈ S_{q}(n, r) : U is upper triangular}.

Proof This follows from [10, Propositions 4.1.3, 4.2.1], because we know by Proposition 3.9 that the two bases are equivalent. □

16 R.M. GREEN

4. The Positivity Result

The next aim is to use Proposition 3.10 to establish the positivity property for S_{q}^{+}(n, r).

Definitions Following [4], we write ϕ̂_{λ,μ}^{w} for v^{ℓ(w^{+})+ℓ(w_{0,μ})}ϕ_{λ,μ}^{w}. Note that if ϕ_{λ,μ}^{w} corresponds to A ∈ M(n, r), then ϕ̂_{λ,μ}^{w} = [A].

In [4, §3.1], a ℤ-linear involution, ¯ , on S_{q}(n, r) is described. This sends v to v^{−1} and fixes all θ_{λ,μ}^{w}.

The algebra U^{+} also has a ℤ-linear automorphism, ¯ , sending v to v^{−1} and fixing all the E_{i}. (No confusion will arise in the similarity of notation as will be seen shortly.)

Lemma 4.1

(a) For all possible λ and μ, the identity

θ_{λ,μ}^{1} = ϕ̂_{λ,μ}^{1}

holds in S_{q}(n, r).

(b) For all 1 ≤ i < n, \overline{π_{r}(E_{i})} = π_{r}(E_{i}).

(c) Let u ∈ U^{+}. Then \overline{π_{r}(u)} = π_{r}(ū). Thus if u ∈ U satisfies ū = u then \overline{π_{r}(u)} = π_{r}(u).

Proof The proof of (a) follows from the expression for the θ basis in terms of the ϕ̂ basis given in [4, §3.8], and the fact that if z^{+} ≤ 1^{+} in the Bruhat order then the distinguished double coset representative of z is 1.

The proof of (b) follows using the definition (as in [6]) of π_{r}(E_{i}) which is of the form

∑_{⟨λ,μ⟩} ϕ̂_{λ,μ}^{1}

for a certain set of pairs ⟨λ, μ⟩.

For (c), observe that (b) shows it is true for the generators E_{i}. Because ¯ is an A-semilinear algebra homomorphism and π_{r} is an A-linear algebra homomorphism, the result follows. □

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 17

Definitions We define A^{+} to be ℤ[v], and A^{−} to be ℤ[v^{−1}].

Denote by L the free A^{−}-module with the Lusztig–Dyer basis L as a basis. There is a canonical projection

π_{L} : L → L/v^{−1}L.

Denote by L′ the free A^{−}-module with {[U] : U is upper triangular} as a basis. There is a canonical projection

π_{L′} : L′ → L/v^{−1}L′.

Following [14, Theorem 42.1.10, Corollary 42.1.13], we can characterise the canonical basis B of U^{+}(sl_{n}) by the following two properties.

1. For each b ∈ B, b̄ = b.

2. For each b ∈ B, π_{L}(b) = π_{L}(b′) for exactly one b′ ∈ L. Using Corollary 2.3 and [4, Theorem 3.9], we can characterise the canonical basis

B′ of S_{q}^{+}(n, r) by the properties 1′. For each b ∈ B′, b̄ = b. 2′. For each b ∈ B′, π_{L′}(b) = π_{L′}(b′) for exactly one b′ = [U] where U is upper

triangular.

Proposition 4.2 The nonzero elements

{[D].π_{r}(b) : b ∈ B, D diagonal, D ∈ M(n, r)}

are precisely the canonical basis elements θ_{U} in S_{q}(n, r). Proof We need to verify properties 1′ and 2′ above. To check 1′, we observe that [D] = θ_{λλ}^{1} for some λ. Hence by Lemma 4.1,

\overline{[D]π_{r}(b)} = \overline{[D]}.\overline{π_{r}(b)}
= [D]π_{r}(b̄)
= [D]π_{r}(b).

We now verify 2′. Property 2 for Lusztig’s canonical basis implies that, by expressing b ∈ B as a linear combination of b′, b′′ ∈ L, we can write

[D]π_{r}(b) = [D]π_{r}(b′) + ∑_{b′′≠b′} c_{b′′}[D]π_{r}(b′′),

where [D].π_{r}(b) is as in the statement of the proposition and the coefficients c_{b′′} lie in v^{−1}A^{−}. Thus by Proposition 3.10,

[D]π_{r}(b) ≡ [D]π_{r}(b′) mod L′/v^{−1}L′.

Suppose that [D]π_{r}(b′) = 0. Then [D]π_{r}(b) ∈ v^{−1}L′. This means that

[D]π_{r}(b) = ∑ c_{U}θ_{U},

where c_{U} ∈ v^{−1}A^{−}. (This follows from property 2′ of the canonical basis of S_{q}^{+}(n, r); the elements θ_{U} are an A^{−}-basis of L′.) Applying ¯ to this equation shows that c_{U} = \overline{c_{U}} for all c_{U}. This forces all the c_{U} to be zero, and [D]π_{r}(b) = 0.

This gives the correspondence required by 2′, completing the proof. □

We can now give a partial result towards the positivity conjecture. Corollary 4.3 The algebra S_{q}^{+}(n, r) equipped with its canonical basis has the positivity property. Proof First note that any canonical basis element θ_{A} has the property that θ_{A} = θ_{D}θ_{A}θ_{D′} for unique diagonal matrices D and D′. Also, [D][D′] = δ_{D,D′}[D]. Thus if [D]π_{r}(b_{1}) and [D′]π_{r}(b_{2}) are both canonical basis elements (as before), then

[D]π_{r}(b_{1})[D′]π_{r}(b_{2}) = \{ \begin{matrix}0 & if ([D]π_{r}(b))[D′] = 0, \\ [D]π_{r}(b_{1}b_{2}) & if ([D]π_{r}(b))[D′] = [D]π_{r}(b). \end{matrix}

We now invoke a deep result of Lusztig, namely the positivity property for Lusztig’s canonical basis ([14, Theorem 14.4.13 (a)]). This shows that

b_{1}b_{2} = ∑_{b′∈B} c_{b′}b′,

where c_{b′} ∈ ℕ[v, v^{−1}]. We deduce that

[D]π_{r}(b_{1})[D′]π_{r}(b_{2}) = ∑_{b′∈B} c_{b′}[D]π_{r}(b′),

thus establishing positivity as required. □

The final stage of the proof is to extend this partial result to the whole algebra S_{q}(n, r). To do this, we use the following result.

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 19

Lemma 4.4 (Du) Let θ_{λμ}^{d}, θ_{μ,ν}^{1} (where d ∈ D_{λμ}) be canonical basis elements for S_{q}(n, r). Suppose W_{ν} ⊆ W_{μ}. Then

θ_{λμ}^{d}.θ_{μν}^{1} = θ_{λν}^{d′}

where d′ is such that W_{λ}d^{+}W_{ν} = W_{λ}d′W^{ν}. In particular, if W_{μ} = W_{ν} then d′ = d.

Similarly,

θ_{νμ}^{1}θ_{μλ}^{d^{−1}} = θ_{νλ}^{d^{′−1}}.

Proof This comes from [5, Lemma 5.2]. □

Definition For a, b ∈ {1, 2, 3} and n > 1 we define

D(n, a, b) = {A ∈ M(3n, r) : A_{i,j} ≠ 0 ⇒ ∃x : i = x + (a − 1)n, j = x + (b − 1)n}.

Example Let a = 1, b = 2. Then a typical element of D(n, a, b) is a matrix

( \begin{matrix}0 & D & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

where D is an n × n diagonal matrix and the zero entries represent n × n diagonal matrices with zero entries.

Definition For i, j ∈ {1, 2, 3}, we define σ_{i,j} ∈ S_{q}(3n, r) by

σ_{i,j} = ∑_{D∈D(n,i,j)}[D].

Note that such a [D] is of the form θ_{λ,μ}^{1} where W_{λ} = W_{μ}.

Lemma 4.5

(a) The injective A-linear map ι : S_{q}(n, r) ↪ S_{q}(3n, r) given by ι(θ_{A}) = θ_{A′} where

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

is a monomorphism of A-algebras.

20 R.M. GREEN

(b) On im(ι), σ_{1,1} is the multiplicative identity, that is, σ_{1,1}ι(θ_{A}) = ι(θ_{A})σ_{1,1} = ι(θ_{A})

(c) For i, j, k ∈ {1, 2, 3}, σ_{i,j}σ_{j,k} = σ_{i,k}. Proof Part (a) follows easily from the definition of the canonical basis. For part (b), observe that ι(1) = σ_{1,1}, where 1 is the identity in S_{q}(n, r). Part (c) is a corollary of Lemma 4.4. □

We are now ready to prove the main result. Theorem 4.6 The structure constants for S_{q}(n, r) with respect to the canonical basis lie in ℕ[v, v^{−1}]. Proof Choose θ_{A}, θ_{B} ∈ S_{q}(n, r). We may assume the product θ_{A}θ_{B} is not zero. We express the structure constants as

θ_{A}θ_{B} = ∑_{C∈M(n,r)} f_{A,B,C}θ_{C}.

Hence by Lemma 4.5 (a),

σ_{1,1}ι(θ_{A})ι(θ_{B})σ_{1,3} = ∑_{C∈M(n,r)} f_{A,B,C}σ_{1,1}ι(θ_{C})σ_{1,3}.

Using Lemma 4.5 (b) and (c) we find that

(σ_{1,1}ι(θ_{A})σ_{1,2})(σ_{2,1}ι(θ_{B})σ_{1,3}) = ∑_{C∈M(n,r)} f_{A,B,C}(σ_{1,1}ι(θ_{C})σ_{1,3}).

Note that by Lemma 4.4, all the expressions in parentheses are canonical basis elements of S_{q}(3n, r). More importantly, the matrix indexing the basis element σ_{1,1}θ_{A}σ_{1,2} is of the form

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

and the matrix indexing the basis element σ_{2,1}θ_{B}σ_{1,3} is of the form

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

The matrices corresponding to the elements σ_{1,1}ι(θ_{C})σ_{1,3} are of the form

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

POSITIVITY PROPERTIES FOR q-SCHUR ALGEBRAS 21

These are all upper triangular canonical basis elements, so lie in S_{q}^{+}(3n, r). Thus by Corollary 4.3, all the f_{A,B,C} lie in ℕ[v, v^{−1}]. □

Corollary 4.7 The structure constants for S_{q}(n, r) with respect to the canonical basis are naturally interpretable in terms of intersection cohomology. Proof The proof of Theorem 4.6 shows that the structure constants are the same as certain structure constants in the algebra S_{q}^{+}(3n, r). The proof of Corollary 4.3 shows that these are the same as certain structure constants arising from Lusztig’s canonical basis (provided they are not zero). It is known from [14, Theorem 14.4.13] that these latter constants have a natural interpretation in the framework of perverse sheaves and intersection cohomology. □

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[5] J. Du, Canonical Bases for Irreducible Representations of Quantum GL_{n}, Bull. London Math. Soc. 24 (1992), 325–334.

[6] J. Du, A Note on Quantized Weyl Reciprocity at Roots of Unity, Algebra Colloquium, to appear.

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22 R.M. GREEN

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