6th NWDR Representation Theory Workshop
Paderborn, 21-22 October, 2005

Programme

All talks in lecture room D2.

Friday, 21 October

14.00 - 15.00 Benson
15.00 - 15.30 Coffee
15.30 - 16.30 Liu
16.45 - 17.45 Olbricht
18.30 - Dinner at "Ratskeller"

Saturday, 22 October

09.30 - 10.30 Buchweitz
10.30 - 11.00 Coffee
11.00 - 12.00 Farnsteiner
12.00 - 13.30 Lunch
13.30 - 14.30 Miemietz
14.30 - 15.00 Coffee
15.00 - 16.00 Schröer
16.15 - 17.15 Reiten

Titles and Abstracts

Dave Benson (Aberdeen): Polynomial and power series invariants of finite groups
Abstract: This is a report on joint work with Peter Webb. Let G be a finite group, k a perfect field of characteristic p, and V a finite dimensional kG-module. We let G act on the power series k[[V]] by linear substitutions and address the question of when the invariant power series k[[V]]^G form a unique factorization domain. For a permutation module for a p-group, the answer is always positive. On the other hand, if G is a cyclic group of order p and V is an indecomposable kG-module of dimension r between 1 and p, the invariant power series form a unique factorization domain if and only if r is equal to 1, 2, p-1 or p. This contradicts a conjecture of Peskin. In contrast, a theorem of Nakajima completely answers the question of when the invariant polynomial functions k[V]^G form a unique factorization domain; for a p-group in characteristic p, the answer is always yes.

Ragnar Buchweitz (Toronto): Hochschild Cohomology and Calculus
Abstract: Recently, the natural algebra homomorphism from Hochschild cohomology of an algebra into the graded centre of its derived category has found some attention, for example, in the theory of support varieties of modules. In this talk we want to explain how noncommutative calculus can be used to study this homomorphism and to bound its image in various Ext groups. The key tool are the Atiyah classes and the Atiyah-Chern character of a (complex) of modules.

Rolf Farnsteiner (Bielefeld): Representation-theoretic support spaces
Abstract: Let G be a finite group scheme over an algebraically closed field k of positive characteristic p. In recent work, E. Friedlander and J. Pevstova have introduced the space of abelian p-points, which generalizes rank varieties for finite groups and infinitesimal groups. In my talk, I will first discuss basic properties of p-points and then provide applications concerning the representation theory of the algebra kG of measures on G.

Yuming Liu (Beijing Normal U.): Stable equivalence of Morita type - from self-injective algebras to general finite dimensional algebras
Abstract: The notion of stable equivalence of Morita type plays a substantial role in the representation theory of block algebras of finite groups, or more generally, of self-injective algebras. In this talk, I will present some new development of stable equivalence of Morita type for general finite dimensional algebras. In particular, I will compare the difference between the stable equivalence of Morita type for self-injective algebras and that for general finite dimensional algebras. This is a joint work with C.C. Xi.

Vanessa Miemietz (Stuttgart/Oxford): On representations of affine Hecke algebras of type B
Abstract: Grojnowski's approach to the representation theory of affine Hecke algebras of type A is applied to type B with unequal parameters to obtain - under certain restrictions on the eigenvalues of the lattice operators - analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type A.

Roland Olbricht (Münster): On degenerations between preprojective modules over wild quivers
Abstract: We consider minimal degenerations between preprojective modules over wild quivers. Asymptotic properties of such degenerations are studied, with respect to codimension and numbers of indecomposable direct summands. We provide families of minimal disjoint degenerations of arbitrary codimension for almost all wild quivers and show that no such examples exist in the remaining cases.

Idun Reiten (Trondheim): Clusters and seeds for acyclic cluster algebras
Abstract: Clusters and seeds are essential ingredients in the definition of cluster algebras by Fomin-Zelevinsky. We consider a seed to be a pair (x,Q) where x is a cluster and Q a finite quiver with no cycles of length at most two. We deal with the case of acyclic cluster algebras, that is, one of the quivers in some seed has no oriented cycles (and we assume "no coefficients"). The aim of this lecture is to discuss the result, based upon work with Buan, Marsh, Todorov, that a seed (x,Q) is determined by the cluster x, as conjectured by Fomin-Zelevinsky. The proof uses the theory of cluster categories and cluster-tilted algebras. An important step is a description of the monomials occuring as denominators for clusters variables as corresponding to indecomposable exceptional modules of a finite dimensional path algebra kQ.

Jan Schröer (Bonn): Cluster monomials are dual semicanonical basis vectors
Abstract: Our aim is to explain the statement in the title of the talk. This is joint work with C. Geiss and B. Leclerc.