LMS Durham Symposium:
Representations of Finite Groups and Related
Algebras
1-11 July 2002
Abstracts
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Dave Benson
Modules with injective cohomology
This talk will describe joint work with Henning Krause. We classify the
injective modules over the Tate cohomology ring of a finite group, and we show
how to construct kG-modules whose Tate cohomology is equal to a given
injective. The construction produces pure injective modules which are (almost)
characterized by their cohomology. We conjecture that these are (up to an
explicitly described degree shift) the infinite dimensional kappa modules of
Benson, Carlson and Rickard. The conjecture is true for groups whose
cohomology is Cohen-Macaulay, but is still open in general. This is related to
a conjectural way of calculating the variety of an infinite dimensional module
by forming an injective resolution of its Tate cohomology.
Cedric Bonnafé
Around the Mackey formula for Lusztig functors
The Mackey formula for Lusztig induction and restriction (for finite reductive
groups) is analogous to the Mackey formula for usual induction and restriction
(for abstract finite groups). But it is not proved in full generality. We
explain in this talk our recent progress in this direction : one of our main
result is that the Mackey formula holds for classical groups.
In this talk, we also explain a related result in modular
representation theory of finite reductive groups : with R. Rouquier,
we proved that some Morita equivalence ("Jordan decomposition")
given by the Lusztig functor associated to a Borel subgroup
does not depend on the choice of the Borel subgroup.
Everett Dade
Reduction theorems for conjectures of Uno and of Navarro
A strengthened form of Uno's new conjecture can be reduced by standard
methods to the case of decorated simple groups. But Navarro's
conjecture requires some new methods and structures for a similar
reduction. I'll try to explain exactly what must be verified for each
simple group in order to prove these conjectures by induction.
Steve Donkin
The higher decomposition numbers for SL(3)
For each pair of integers λ and i=0,1,2,3 there is a naturally occuring
module Hi(λ) for the algebraic group SL(3), with coefficients
in an algebraically closed field of positive characteristic p. (The cohomology
group of the line bundle Lλ on the flag variety G/B.) The
modules H0(λ) are the duals of the Weyl modules. The
irreducible modules for the algebraic group SL(3) are well known and are
indexed by pairs μ of nonnegative integers. The composition multiplicities
of the Weyl modules are also well known. We here describe the composition
multiplicities [Hi(λ):L(μ)], for all i,λ,μ, in
terms of the base p-expansion of λ. The proof involves some of the
partial tilting modules for SL(3).
Robert Hartmann
Endo-monomial modules
For a complete discrete valuation ring O and a p-group P,
we generalize the concept of endo-permutation modules by considering
OP-modules whose endomorphism algebra forms a monomial OP-module. We
introduce the analog of the Dade group and show that for abelian P every
indecomposable endo-monomial OP-module with vertex P is already an
endo-permutation module.
Frank Himstedt
On the decomposition numbers of Steinberg's triality groups
In 1991, M. Geck has determined the decomposition numbers of Steinberg's
triality groups 3D4(q), q odd, in characteristics not
dividing q, leaving some ambiguities in the decomposition numbers of the
unipotent characters.
In this talk, new results are presented about one of these ambiguities, which
have been obtained using techniques similar to those introduced by T. Okuyama
and K. Waki in their determination of the decomposition numbers of the
symplectic groups Sp(4,q) in 1996 and 1998.
First, using computers, the character table of a maximal parabolic subgroup of
3D4(q) has been calculated. Then, an analysis of
restrictions of suitable modules from 3D4(q) to this
subgroup using Green correspondence has given new information on one of the
ambiguities in the decomposition numbers of the unipotent characters.
Thorsten Holm
Auslander's representation dimension
(joint work with K. Erdmann and J. Schröer)
M. Auslander introduced the representation dimension of an algebra around 1970
as a possible way of measuring how far an algebra is from being of finite
representation type. In fact, he proved that being representation finite is
equivalent to having representation dimension 2. Apart from this result the
notion remained mysterious and only very recently new progress
emerged. Firstly, O. Iyama showed that the representation dimension is always
finite. This is obtained by showing that there exists a certain module having
quasi-hereditary endomorphism ring. Secondly, C. Xi proved that for
selfinjective algebras the representation dimension is invariant under derived
equivalence.
In the talk we report on recent progress on the representation dimension for
group algebras and related algebras. Two main results will be discussed:
1) All special biserial algebras (e.g. blocks
of finite groups with dihedral defect group) have representation
dimension 3.
2) All blocks of finite groups with tame representation
type (and more generally, all algebras of dihedral, semidihedral
and quaternion type) have representation dimension 3.
Shigeo Koshitani
On Broué's abelian defect group conjecture
There is a well-known and important conjecture called "Broué's abelian
defect group conjecture". By this we here mean his conjecture on splendid
Rickard equivalences. We are going to discuss by taking some examples of
non-principal blocks. Hopefully, some of them should be new.
Gunter Malle
Springer correspondence for disconnected groups
The Springer correspondence is an injective map from the set of unipotent
classes of an algebraic group to the set of characters of its Weyl group. It
plays a crucial role in Lusztig's algorithm for the computation of character
values of finite groups of Lie type. In her PhD thesis, Karin Sorlin has
extended the Springer correspondence to the case of disconnected groups. In
joint work we have now determined this correspondence in the case of
disconnected classical groups. It is expected that this will eventually allow
to compute character values for these disconnected groups.
Hyohe Miyachi
Uno's conjecture on 1-parameter Iwahori-Hecke algebras
In the early 90's, K.Uno determined the representation type of the
(1-parameter) Iwahori-Hecke algebras for the Coxeter groups with rank 2 and the
finite Weyl groups of type A. Uno conjectured that a given Iwahori-Hecke
algebra H is not semisimple and of finite type if and only if the parameter of
H is a simple root of the Poincare polynomial for the Weyl group of
H. Recently, by Ariki-Mathas (math.RT/0106185) and Ariki
(math.QA/0108176) we can know all the representation types of classical
(1-parameter) Hecke algebras. Motivated by these works, we determine the
representation type of the (1-parameter) Iwahori-Hecke algebras for the
(crystallographic) exceptional Weyl groups over a splitting field with a good
characteristic.
Max Neunhöffer
Idempotents in symmetric algebras
Let H be a non-semisimple symmetric algebra over a field K. In this talk
explicit formulae for primitive idempotents of H are presented, which involve
only the matrix coefficients of certain matrix representations corresponding to
projective indecomposable modules.
Christakis Pallikaros
Kazhdan-Lusztig cells
and parabolic elements in finite Coxeter groups
The Hecke algebra H of a Coxeter group W has certain ideals associated with the
longest elements of the parabolic subgroups of W which allow a straightforward
analysis of the corresponding cell representations. These ideals have the form
CwH where C refers to the C-basis introduced by Kazhdan and
Lusztig. We show that the isomorphism classes of these ideals depend only on
the Coxeter classes of the corresponding parabolic subgroups, as do the 2-sided
cells containing their longest elements. In the special case of Hecke algebras
of type A, we show that this correspondence is bijective.
Leonard Scott
Some empirical and empirically-inspired results
in Lie-type modular representation theory
I will discuss a constructive, though recursive, construction of the radical
quotient of the maximal submodule of a standard (Verma, Weyl, baby Verma, ...)
module, in the presence of a valid Kazhdan-Lusztig or Lusztig conjecture. The
result was found empirically, though verified theoretically, and I will discuss
some of the related computer calculations. Some of these calculations led to
verification of new small cases of the Lusztig conjecture in characteristic p,
while others led to counterexamples to a conjecture of some standing on
dimensions of 1-cohomology groups for finite groups with faithful absolutely
irreducible coefficients.
Nicole Snashall
Support varieties and Hochschild cohomology rings
This is joint work with Ø. Solberg.
We define a support variety for a finitely generated module over any artin
algebra Λ in terms of the maximal ideal spectrum of the Hochschild
cohomology ring of Λ. This is modelled on what is done in modular
representation theory, and the varieties defined in this way are shown to have
many of the same properties as for group rings. Information is also given on
nilpotent elements of HH*(Λ) and I will discuss the question
of whether the quotient algebra HH*(Λ) modulo the ideal
generated by homogeneous nilpotent elements is finitely generated as an
algebra.
Kai Meng Tan
On Rouquier blocks of symmetric groups and Schur algebras
I will present explicit formulas for the radical layers of the principal
indecomposable, Weyl, Young and Specht modules of these blocks, and address
some conjectures of James, Martin, Lascoux-Leclerc-Thibon-Rouquier for these
blocks.
Will Turner
RoCK blocks of symmetric groups of nonabelian defect
Rouquier has defined a family of symmetric group blocks which possess a very
symmetric structure. Chuang and Kessar have proved, for example, that these
blocks are Morita equivalent to principal blocks of wreath products
Spwr Sw of symmetric groups, so long as their defect
group is abelian. I would like to describe some theory for "RoCK" blocks of
nonabelian defect. For example, I shall state a formula for the decomposition
matrices of these blocks, given in terms of Littlewood-Richardson coefficients
and decomposition numbers of "small" Schur algebras. This generalises a formula
obtained by Chuang and Tan in the abelian defect case.
Katsuhiro Uno
Modifications and applications of reduction theorems for conjectures
on character degrees
Let G be a finite group and p a prime. We consider a conjecture on the
alternating sum of the numbers of complex irreducible characters in certain
blocks of chain normalizers having fixed defect and ± fixed residue mod
p. Here, the defect d of a character χ of a subgroup H of G is defined by
(|H|/χ(1))p = pd and its residue is defined by
(|H|/χ(1))p'. This conjecture is a refinement of Dade's, and
moreover, its projective form implies the Isaacs-Navarro conjecture, just like
that form of Dade's implies the Alperin-McKay. Dade proved a reduction theorem
for the projective form of his conjecture, and it turns out that it is also
valid for ours. We give some modifications of it and apply them in several
cases. In particular, finite groups with small Sylow p-subgroups and sporadic
simple groups will be treated.
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