Preprints and publications

Joint with T. Halverson. We compute the dimension d_n,r(q) = dim(IR_q^r) of the defining module IR_q^r for the q-partition algebra. This module comes from r-iterations of Harish-Chandra restriction and induction on GL_n(F_q). This dimension is a polynomial in q that specializes as d_n,r(1) = n^r and d_n,r(0) = B(r), the rth Bell number. We compute d_n,r(q) in two ways. The first is purely combinatorial. We show that d_n,r(q) = ∑_λ f ^λ(q) m_r^λ, where f ^λ(q) is the q-hook number and m_r^λ is the number of r-vacillating tableaux. Using a Schensted bijection, we write this as a sum over integer sequences which, when q-counted by inverse major index, gives d_n,r(q). The second way is algebraic. We find a basis of IR_q^r that is indexed by n-restricted q-set partitions of {1,..., r}, and we show that there are d_n,r(q) of these. pdf (2008).

Superinduction for pattern groups

Joint with E. Marberg. pdf (2007).

Restricting supercharacters of the finite group of unipotent uppertriangular matrices

Joint with V. Venkateswaran. pdf (2007).

It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices U_n over a finite field is a wild problem. By instead considering approximately irreducible representations, one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. These two papers explore two different aspects of this connection. The first paper studies superinduction, and then gives an explicit combinatorial algorithm for computing induction in the case U_n in a way that is analogous to the Pieri-formula for the symmetric group. The second paper studies a family of subgroups that interpolate between U_n-1 and U_n, and uses supercharacter theoretic results to compute combinatorial restriction formulas for U_n. Together these papers suggest that there should be an invariant theory for U_n analogous to the ring of symmetric functions for the symmetric group.

Joint with C.R. Vinroot. A known result for the finite general linear group GL(n,F_q) and for the finite unitary group U(n,F_q²) posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman and Guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of GL(n,F_q). We develop an explicit formula for the value of the permutation character of U(2n,F_q²) over Sp(2n,F_q) evaluated an an arbitrary conjugacy class and use results concerning Gelfand-Graev characters to obtain an analogous formula for U(n,F_q²) in the case where q is an odd prime power. These results are also given as probabilistic statements. pdf. To appear in J. Algebra.

Joint with C.R. Vinroot. Gelfand-Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand-Graev characters of the finite unitary group from a combinatorial point of view. In particular, we give the values of Gelfand-Graev characters at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand-Graev characters in terms of tableau combinatorics, and conclude with some multiplicity consequences. pdf.

Joint with P. Diaconis. C. Andre and N. Yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group U_n(F_q). In this theory superclasses are certain unions of conjugacy classes, and supercharacters are a set of characters which are constant on superclasses. This paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups. .pdf . To appear in the Transactions of the American Mathematical Society.

Joint with C.R. Vinroot. This paper describes the analogue of Green's characteristic map for the unitary group, and gives some representation theoretic applications. Advances in Mathematics 210 (2007): 707-732. pdf .

This paper extends the work of the paper below by analyzing the\ multiplication of basis elements in Unipotent Hecke algebras. This work is done in general type, but a section is devoted to working out the combinatorics in the general linear group case. Transactions of the American Mathematical Society 359 (2007): 1685-1724. pdf .

This is my thesis pdf .