Représentations des groupes finis et des groupes algébriques
Org: Gerald Cliff (Alberta) et Anna Stokke (Winnipeg) [PDF]
 PETER CAMPBELL, University of Bristol, Bristol, U.K.
Ktypes for principal series representations of GL(3)
[PDF] 
Let o be the ring of integers of a padic field F,
then K = GL(3,o) is a maximal compact subgroup of the
padic group G = GL(3,F). On restriction to K, a principal
series representation of G necessarily decomposes as the direct sum
of smooth irreducible representations of K each appearing with
finite multiplicity. We will give a description of this
decomposition, with particular emphasis on the unramified case, and
outline an application to an analogue of the Steinberg representation
for the finite group GL(n,o/p^{l}) where
p is the prime ideal of o.
This is joint work with Monica Nevins (University of Ottawa).
 XUEQING CHEN, University of WisconsinWhitewater
Quivers, RingelHall Algebras and Lie Theory
[PDF] 
Let g = g(C) be the KacMoody Lie algebra associated to a Cartan
matrix C and U = U_{v} (g) its quantum group. A
key feature in quantum groups is the presence of several natural bases
(like the PBWbasis and the canonical basis). There are different
approaches to the construction of the canonical basis: algebraic
approach, geometric approach and RingelHall algebra approach. In
this talk, we start by recalling the basic theory of quivers and
RingelHall algebras, paying special attention to Gabriel's Theorem
and RingelGreen's work on the realization of quantum groups and Lie
algebras by using Hall algebras of finite dimensional associative
algebras. We will then recall algebraic and RingelHall algebra
approaches to a PBW basis and a canonical basis of U when
C is of finite or affine type. Meanwhile, the root vectors in
RingelHall algebras will be discussed. Finally, we shall go on to
discuss some of the many further developments and applications of the
theory.
 JAYDEEP CHIPALKATTI, University of Manitoba
The higher transvectants are redundant
[PDF] 
Transvectants were discovered in the nineteenth century by the
German school of classical invariant theorists. In modern
terminology, they encode the decomposition of the tensor product of
two SL_{2}representations. Given binary forms A
and B of orders d,e respectively, their rth transvectant T_{r} = (A,B)_{r} is a form of order d+e2r whose coefficients are
bilinear in those of A,B. We classify all quadratic syzygies in
the T_{r}, and show that as a consequence, the higher transvectants
{T_{r}}_{r ³ 2} can be entirely recovered from T_{0},T_{1}.
This is joint work with A. Abdesselam from Université Paris
XIII.
 ALLEN HERMAN, University of Regina, Regina, SK, S4S 0A2
The Group of Ring Automorphisms of a Rational Group Algebra
[PDF] 
Ring automorphisms of the rational group algebra QG of a
finite group G come in two types: inner automorphisms that leave
every simple component of QG invariant, and outer
automorphisms that interchange at least two of these simple
components.
In order to detect the existence of outer automorphisms of
QG, one must be able to determine whether or not two simple
components of QG are ring isomorphic. It is automatic that
two isomorphic simple components will have isomorphic centers, equal
dimension, and the same local Schur indices at each rational prime.
However, an example of a group will be given to show that these
conditions are not sufficient. It has two simple components that are
ring isomorphic but not Morita equivalent. Procedures for determining
whether two simple components of QG are ring isomorphic
will be discussed.
 GREG LEE, Lakehead University, Thunder Bay, Ontario
Nilpotent symmetric units in group rings
[PDF] 
The group ring KG of a group G over a field K has a natural
involution * sending each group element to its inverse. The
elements fixed by this involution are said to be symmetric.
We will examine the symmetric units in KG, and discuss the
conditions under which they satisfy a group identity. In
particular, we will explore the conditions under which the symmetric
units are nilpotent. Until recently, all of the known results
concerned torsion groups. However, new results allow us to consider
groups with elements of infinite order.
 DAVID MCNEILLY, Alberta

 FERNANDO SZECHTMAN, University of Regina
Modular Reduction of the Steinberg Lattice of the General
Linear Group
[PDF] 
The mod l reduction of the Steinberg lattice of G = GL(n,q) is
known not to be irreducible when l divides the index [G:B] of
the upper triangular group B in G. In 2003 R. Gow produced a
canonical filtration for this reduced module, and conjectured that all
nonzero factors are irreducible. The bottom factor is the socle,
which has been known to be irreducible for some time. We consider the
next factor of Gow's filtration, lying just above the socle, and prove
it to be irreducible in the particular case when l divides q+1.
 QIANGLONG WEN, University of Alberta, 114 St  89 Ave, Edmonton, Alberta,
Canada, T6G 2E1
Some Irreducible Characters of GL(2,Z/p^{n}Z) and
GL(3,Z/p^{n}Z)
[PDF] 
Nowadays there is considerable interest in the representations of
GL(n,Z_{p}), where GL(n,Z_{p}) are the padic integers. Since
every continuous irreducible representation of GL(n,Z_{p}) comes
from a representation of GL(n,Z_{p}/p^{m}Z_{p}) and Z_{p}/p^{m}Z_{p}) @ Z/p^{m}Z, I focus on finding some irreducible characters of
GL(n,Z/p^{m}Z). Clifford Theory gives us a method to construct
irreducible characters of a group G, by inducing up certain
irreducible characters of subgroups H of G. I apply Clifford
Theory to construct three types of irreducible characters of groups
GL(2,Z/p^{n}Z) and GL(3,Z/p^{n}Z).
