Past Representation Theory Seminar

In this talk I will discuss how to construct generalized traces

and modified dimensions in certain categories of modules. As I will explain

there are several examples in representation theory where the usual trace

and dimension are zero, but these generalized traces and modified dimensions

are non-zero. Such examples include the representation theory of the Lie

algebra sl(2) over a field of positive characteristic and of Lie

superalgebras over the complex numbers. In these examples the modified

dimensions can be interpreted categorically and are closely related to some

basic notions involving the representation theory. This joint work with Jon

Kujawa and Bertrand Patureau.

This is joint work with Diaz, Glesser and Park.

In Proc. Instructional Conf, Oxford 1969, G. Glauberman shows that

several global properties of a finite group are determined by the properties

of its p-local subgroups for some prime p. With Diaz, Glesser and Park, we

reviewed these results by replacing the group by a saturated fusion system

and proved that the ad hoc statements hold. In this talk, we will present

the adapted versions of some of Glauberman and Thompson theorems.

In this talk we present a new construction of a Wedderburn basis for

the generic q-Schur algebra using the Du-Kazhdan-Lusztig basis. We show

that this gives rise to a new view on the Du-Lusztig homomorphism to the

asymptotic algebra. At the end we explain a potential plan for an attack

on James' conjecture using a reformulation by Meinolf Geck.

The talk starts with a gentle recollection of facts about

Iwahori-Hecke-Algebras of type A and q-Schur algebras and aims to be

accessible to people who are not (yet) experts in the representation

theory of q-Schur algebras.

All this is joint work with Olivier Brunat (Bochum).

It is well-known that there is a strong link between the representation

theories of the general linear group and the symmetric group over the

complex numbers. J.A.Green has shown that this in also true over infinite

fields of positive characteristic. For this he used the Schur functor as

introduced by I.Schur in his PhD thesis.

In this talk I will show that one can do the same thing for the symplectic

group and the Brauer algebra. This is joint work with S.Donkin. As a

consequence we obtain that (under certain conditions) the Brauer algebra and

the symplectic Schur algebra in characteristic p have the same block

relation. Furthermore we obtain a new proof of the description of the blocks

of the Brauer algebra in characteristic zero as obtained by Cox, De Visscher

and Martin.

We consider the symmetric group S_n of degree n and an algebraically

closed field F of prime characteristic p.

As is well-known, many representation theoretical objects of S_n

possess concrete combinatorial descriptions such as the simple

FS_n-modules through their parametrization by the p-regular partitions of n,

or the blocks of FS_n through their characterization in terms of p-cores

and p-weights. In contrast, though closely related to blocks and their

defect groups, the vertices of the simple FS_n-modules are rather poorly

understood. Currently one is far from knowing what these vertices look

like in general and whether they could be characterized combinatorially

as well.

In this talk I will refer to some theoretical and computational

approaches towards the determination of vertices of simple FS_n-modules.

Moreover, I will present some results concerning the vertices of

certain classes of simple FS_n-modules such as the ones labelled by

hook partitions or two part partitions, and will state a series of

general open questions and conjectures.

In 1989, Happel raised the following question: if the Hochschild cohomology

groups of a finite dimensional algebra vanish in high degrees, then does the

algebra have finite global dimension? This was answered negatively in a

paper by Buchweitz, Green, Madsen and Solberg. However, the Hochschild

homology version of Happel's question, a conjecture given by Han, is open.

We give a positive answer to this conjecture for local graded algebras,

Koszul algebras and cellular algebras. The proof uses Igusa's formula for

relating the Euler characteristic of relative cyclic homology to the graded

Cartan determinant. This is joint work with Dag Madsen.

joint work with R Gurjar

For a positively graded algebra A we construct a functor from the derived

category of graded A-modules to the derived category of graded modules over

the quadratic dual A^! of A. This functor is an equivalence of certain

bounded subcategories if and only if the algebra A is Koszul. In the latter

case the functor gives the classical Koszul duality. The approach I will

talk about uses the category of linear complexes of projective A-modules.

Its advantage is that the Koszul duality functor is given in a nice and

explicit way for computational applications. The applications I am going to

discuss are Koszul dualities between certain functors on the regular block

of the category O, which lead to connections between different

categorifications of certain knot invariants. (Joint work with S.Ovsienko

and C.Stroppel.)

Lusztig discover an integral lift of the Frobenius morphism for algebraic groups in positive characteristic to quantum groups at a root of unity. We will describe how this map may be constructed via the Hall algebra realization of a quantum group.