Preprints and publications

Hopf monoids from class functions on unitriangular matrices

Joint with Marcelo Aguiar and Nantel Bergeron. We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices. pdf (2012).

A supercharacter table decomposition via power-sum symmetric functions

Joint with Nantel Bergeron. We give an AB-factorization of the supercharacter table of the group of nx n unipotent upper triangular matrices over Fq, where A is a lower-triangular matrix with entries in Z[q] and B is a unipotent upper-triangular matrix with entries in Z[q-1]. To this end we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommutative variables. The factorization is obtain from the transition matrices between the supercharacter basis, the $q$-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group Sn given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular we compute the determinant of the supercharacter table. pdf (2011).

The negative q-binomial

Joint with Shishuo Fu, Vic Reiner, and Dennis Stanton. Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces. pdf Electronic Journal of Combinatorics 19(1) P36 (2012), 24 pages.

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Joint with Marcelo Aguiar , Carlos Andre, Carolina Benedetti, Nantel Bergeron , Zhi Chen, P. Diaconis, Anders Hendrickson, Samuel Hsiao, I. Martin Isaacs, Andrea Jedwab, Kenneth Johnson, Gizem Karaali, Aaron Lauve, Tung Le, S. Lewis, Huilan Li, Kay Magaard, Eric Marberg, Jean-Christophe Novelli, Amy Pang, Franco Saliola, Lenny Tevlin, Jean-Yves Thibon, Vidya Venkateswaran, C.R. Vinroot, Ning Yan, Mike Zabrocki.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. pdf, Advances in Mathematics 229 (2012), 2310-2337.

Nonzero coefficients in restrictions and tensor products of supercharacters of Un(q)

Joint with S. Lewis. The standard supercharacter theory of the finite unipotent upper-triangular matrices Un(q) gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of Um(q)Un(q) for m ≤ n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of Un(q) is a nonnegative integer linear combination of supercharacters of Um(q) (in fact, it is polynomial in q). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of Un(q), this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs. Advances in Mathematics 227 (2011), 40-72. pdf.

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context. Journal of Algebraic Combinatorics 31 (2010), 267-298, pdf.

q-Partition Algebra Combinatorics

Joint with T. Halverson. We compute the dimension dn,r(q) = dim(IRqr) of the defining module IRqr for the q-partition algebra. This module comes from r-iterations of Harish-Chandra restriction and induction on GLn(Fq). This dimension is a polynomial in q that specializes as dn,r(1) = nr and dn,r(0) = B(r), the rth Bell number. We compute dn,r(q) in two ways. The first is purely combinatorial. We show that dn,r(q) = ∑λ f λ(q) mrλ, where f λ(q) is the q-hook number and mrλ 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 dn,r(q). The second way is algebraic. We find a basis of IRqr that is indexed by n-restricted q-set partitions of {1,..., r}, and we show that there are dn,r(q) of these. Journal of Combinatorial Theory Series A 117 (2010), 507-527, pdf.

Superinduction for pattern groups

Joint with Eric Marberg. Journal of Algebra 321 (2009), 3681-3703. pdf.

Restricting supercharacters of the finite group of unipotent uppertriangular matrices

Joint with V. Venkateswaran. pdf Electronic Journal of Combinatorics 16(1) Research paper 23 (2009), 32 pages.

It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices Un 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 Un 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 Un-1 and Un, and uses supercharacter theoretic results to compute combinatorial restriction formulas for Un. Together these papers suggest that there should be an invariant theory for Un analogous to the ring of symmetric functions for the symmetric group.

Values of characters sums for finite unitary groups

Joint with C.R. Vinroot. A known result for the finite general linear group GL(n,Fq) and for the finite unitary group U(n,Fq2) 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,Fq). We develop an explicit formula for the value of the permutation character of U(2n,Fq2) over Sp(2n,Fq) evaluated an an arbitrary conjugacy class and use results concerning Gelfand-Graev characters to obtain an analogous formula for U(n,Fq2) in the case where q is an odd prime power. These results are also given as probabilistic statements. J. Algebra 320 (2008), 1150-1173. pdf.

Gelfand-Graev characters of the finite unitary groups

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. Electronic Journal of Combinatorics 16(1) Research Paper 146 (2009), 37 pages.

Supercharacter formulas for pattern groups

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 Un(Fq). 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. Transactions of the American Mathematical Society 361 (2009), 3501-3533.

On the characteristic map of finite unitary groups

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 .

A skein-like multiplication algorithm for unipotent Hecke algebras

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 .

Unipotent Hecke algebras of GLn(Fq)

This paper examines a family of Hecke algebras that generalize both the Iwahori-Hecke algebra and the Gelfand-Graev Hecke algebra. I give an explicit basis for these algebras, describe a Cartan-like subalgebra and allow the representation theory to inspire a generalization of the RSK correspondence. Journal of Algebra 284 (2005): 559-577.

Unipotent Hecke algebras: the structure, representation theory and combinatorics

This is my thesis pdf .

Quadratic Corestriction, C2-embedding problems, and explicit construction

Joint with J. Swallow. This paper came out of an REU at Davidson College. We investigate aspects of the inverse Galois problem. Communications in Algebra 30 (2002): 3227-3258.