45 research outputs found
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.Comment: 24 page
Supercharacter theories of type unipotent radicals and unipotent polytopes
Even with the introduction of supercharacter theories, the representation
theory of many unipotent groups remains mysterious. This paper constructs a
family of supercharacter theories for normal pattern groups in a way that
exhibit many of the combinatorial properties of the set partition combinatorics
of the full uni-triangular groups, including combinatorial indexing sets,
dimensions, and computable character formulas. Associated with these
supercharacter theories is also a family of polytopes whose integer lattice
points give the theories geometric underpinnings
Supercharacter formulas for pattern groups
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 . 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
Restricting supercharacters of the finite group of unipotent uppertriangular matrices
It is well-known that the representation theory of the finite group of
unipotent upper-triangular matrices over a finite field is a wild
problem. By instead considering approximately irreducible representations
(supercharacters), one obtains a rich combinatorial theory analogous to that of
the symmetric group, where we replace partition combinatorics with
set-partitions. This paper studies the supercharacter theory of a family of
subgroups that interpolate between and . We supply several
combinatorial indexing sets for the supercharacters, supercharacter formulas
for these indexing sets, and a combinatorial rule for restricting
supercharacters from one group to another. A consequence of this analysis is a
Pieri-like restriction rule from to that can be described on
set-partitions (analogous to the corresponding symmetric group rule on
partitions)
The combinatorics of generalized Gelfand--Graev characters
Introduced by Kawanaka in order to find the unipotent representations of
finite groups of Lie type, generalized Gelfand--Graev characters have remained
somewhat mysterious. Even in the case of the finite general linear groups, the
combinatorics of their decompositions has not been worked out. This paper
re-interprets Kawanaka's definition in type in a way that gives far more
flexibility in computations. We use these alternate constructions to show how
to obtain generalized Gelfand--Graev representations directly from the maximal
unipotent subgroups. We also explicitly decompose the corresponding generalized
Gelfand--Graev characters in terms of unipotent representations, thereby
recovering the Kostka--Foulkes polynomials as multiplicities
Restrictions of rainbow supercharacters
The maximal subgroup of unipotent upper-triangular matrices of the finite
general linear groups are a fundamental family of -groups. Their
representation theory is well-known to be wild, but there is a standard
supercharacter theory, replacing irreducible representations by
super-representations, that gives us some control over its representation
theory. While this theory has a beautiful underlying combinatorics built on set
partitions, the structure constants of restricted super-representations remain
mysterious. This paper proposes a new approach to solving the restriction
problem by constructing natural intermediate modules that help "factor" the
computation of the structure constants. We illustrate the technique by solving
the problem completely in the case of rainbow supercharacters (and some
generalizations). Along the way we introduce a new -analogue of the binomial
coefficients that depend on an underlying poset
Nonzero coefficients in restrictions and tensor products of supercharacters of
The standard supercharacter theory of the finite unipotent upper-triangular
matrices gives rise to a beautiful combinatorics based on set
partitions. As with the representation theory of the symmetric group,
embeddings of for lead to branching rules.
Diaconis and Isaacs established that the restriction of a supercharacter of
is a nonnegative integer linear combination of supercharacters of
(in fact, it is polynomial in ). In a first step towards
understanding the combinatorics of coefficients in the branching rules of the
supercharacters of , 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.Comment: 28 page
Gelfand-Graev characters of the finite unitary groups
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
On the characteristic map of finite unitary groups
In his classic book on symmetric functions, Macdonald describes a remarkable
result by Green relating the character theory of the finite general linear
group to transition matrices between bases of symmetric functions. This
connection allows us to analyze the representation theory of the general linear
group via symmetric group combinatorics. Using the work of Ennola, Kawanaka,
Lusztig and Srinivasan, this paper describes the analogous setting for the
finite unitary group. In particular, we explain the connection between
Deligne-Lusztig theory and Ennola's efforts to generalize Green's work, and
deduce various representation theoretic results from these results.
Applications include finding certain sums of character degrees, and a model of
Deligne-Lusztig type for the finite unitary group, which parallels results of
Klyachko and Inglis and Saxl for the finite general linear group
Values of characters sums for finite unitary groups
A known result for the finite general linear group \GL(n,\FF_q) and for the
finite unitary group \U(n,\FF_{q^2}) 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,\FF_q). We
develop an explicit formula for the value of the permutation character of
\U(2n,\FF_{q^2}) over \Sp(2n,\FF_q) evaluated an an arbitrary conjugacy
class and use results concerning Gelfand-Graev characters to obtain an
analogous formula for \U(n,\FF_{q^2}) in the case where is an odd prime.
These results are also given as probabilistic statements