5 research outputs found
Heisenberg characters, unitriangular groups, and Fibonacci numbers
Let \UT_n(\FF_q) denote the group of unipotent upper triangular
matrices over a finite field with elements. We show that the Heisenberg
characters of \UT_{n+1}(\FF_q) are indexed by lattice paths from the origin
to the line using the steps , which are
labeled in a certain way by nonzero elements of \FF_q. In particular, we
prove for that the number of Heisenberg characters of
\UT_{n+1}(\FF_q) is a polynomial in with nonnegative integer
coefficients and degree , whose leading coefficient is the th Fibonacci
number. Similarly, we find that the number of Heisenberg supercharacters of
\UT_n(\FF_q) is a polynomial in whose coefficients are Delannoy numbers
and whose values give a -analogue for the Pell numbers. By counting the
fixed points of the action of a certain group of linear characters, we prove
that the numbers of supercharacters, irreducible supercharacters, Heisenberg
supercharacters, and Heisenberg characters of the subgroup of \UT_n(\FF_q)
consisting of matrices whose superdiagonal entries sum to zero are likewise all
polynomials in with nonnegative integer coefficients.Comment: 25 pages; v2: material significantly revised and condensed; v3: minor
corrections, final versio
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
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. (C) 2012 Elsevier Inc. All rights reserved
The combinatorics of normal subgroups in the unipotent upper triangular group
Describing the conjugacy classes of the unipotent upper triangular groups
uniformly (for all or many values of and
) is a near impossible task. This paper takes on the related problem of
describing the normal subgroups of . For a
prime, a bijection will be established between these subgroups and pairs of
combinatorial objects with labels from . Each pair
comprises a loopless binary matroid and a tight splice, an apparently new kind
of combinatorial object which interpolates between nonnesting partitions and
shortened polyominoes. For arbitrary , the same approach describes a natural
subset of normal subgroups: those which correspond to the ideals of the Lie
algebra under an approximation of the
exponential map.Comment: 31 pages, 1 figur
On the characters of the Sylow p-subgroups of untwisted Chevalley groups Y_n(p^a)
Let be a Sylow p-subgroup of an untwisted Chevalley group
of rank n defined over where q is a power of a prime p. We
partition the set of irreducible characters of into
families indexed by antichains of positive roots of the root system of type
. We focus our attention on the families of characters of which
are indexed by antichains of length 1. Then for each positive root we
establish a one to one correspondence between the minimal degree members of the
family indexed by and the linear characters of a certain subquotient
of . For our single root character
construction recovers amongst other things the elementary supercharacters of
these groups. Most importantly though this paper lays the groundwork for our
classification of the elements of , and