## Preprints and publications |
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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 A is a lower-triangular matrix with entries in Z[q]
and B is a unipotent upper-triangular matrix with entries in
Z[q. To this end we introduce a ^{-1}]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 S 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).
_{n}The negative q-binomial
Joint with Shishuo Fu, Vic
Reiner, and Dennis
Stanton.
Interpretations for the
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, Nonzero coefficients in restrictions and tensor products of supercharacters of U
_{n}(q) Joint with S. Lewis. The standard supercharacter theory of the
finite unipotent upper-triangular matrices U ⊆ _{m}(q)U for _{n}(q)m ≤
n lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of U
is a nonnegative integer linear combination of supercharacters of
_{n}(q)U (in fact, it is polynomial in _{m}(q)q). In a first step towards understanding
the combinatorics of coefficients in the branching rules of the supercharacters of U, 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.
_{n}(q)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.
q-Partition Algebra Combinatorics
Joint with T.
Halverson. We compute the dimension (IR of the defining module
_{q}^{r})IR for the _{q}^{r}q-partition algebra.
This module comes from r-iterations of Harish-Chandra restriction
and induction on GL.
This dimension is a polynomial in _{n}(F_{q})q that specializes as
d and _{n,r}(1) = n^{r}d, the _{n,r}(0) =
B(r)rth Bell number.
We compute d in two ways. The first is purely
combinatorial. We show that _{n,r}(q)d, where
_{n,r}(q) = ∑_{λ
}f^{ λ}(q) m_{r}^{λ}f is the ^{ λ}(q)q-hook number and
m is the number of _{r}^{λ}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.
The second way is algebraic. We find a basis of
_{n,r}(q)IR that is indexed by _{q}^{r}n-restricted
q-set partitions of {1,..., r}, and we show that there are
d of these.
_{n,r}(q)Journal of Combinatorial Theory Series A 117 (2010),
507-527, pdf.
Superinduction for pattern groups Joint with Eric Marberg. Restricting supercharacters of the finite group of unipotent uppertriangular matrices Joint with V. Venkateswaran.
pdf It is well-known that the representation theory of the finite group of unipotent
upper-triangular matrices U 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
_{n}U and _{n-1}U, and uses
supercharacter theoretic results to compute combinatorial
restriction formulas for _{n}U. Together these papers
suggest that there should be an invariant theory for _{n}U
analogous to the ring of symmetric functions for the symmetric group. _{n}Values of characters sums for finite unitary groups Joint with C.R.
Vinroot. A known result for the finite general linear
group GL (n,F 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_{q2})(n,F. We develop an explicit
formula for the value of
the permutation character of U_{q})(2n,F over
Sp_{q2})(2n,F evaluated an an
arbitrary conjugacy
class and use results concerning Gelfand-Graev characters to obtain an
analogous formula for U_{q})(n,F
in the case where _{q2})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.
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
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.
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.
Unipotent Hecke algebras of GL _{n}(F_{q})
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. Unipotent Hecke algebras: the structure, representation theory and combinatorics Quadratic Corestriction, C-embedding problems, and explicit
construction
_{2} Joint with J. Swallow. This paper came out of an REU at Davidson
College. We investigate aspects of the inverse Galois problem. |