NWDR Workshop Winter 2016 – Abstracts

Christine Bessenrodt (Hannover)
Kronecker products of characters of the symmetric groups and their double covers

Decomposing Kronecker products of irreducible characters of the symmetric groups (or equivalently, of inner products of Schur functions) is a longstanding central problem in representation theory and algebraic combinatorics. The talk will focus on special Kronecker products and related problems for skew characters, in particular on the recent classification of multiplicity-free Kronecker products of irreducible characters of the symmetric groups, conjectured in 1999. Also related conjectures and results on spin characters of the double cover groups will be discussed, and the connection between them will be illustrated by some applications of spin characters towards results for symmetric groups.

Gabor Elek (Lancaster)
Convergence and limits of finite dimensional representations of algebras

Motivated by the limit theory of finite graphs I will introduce the notion of metric convergence of finite dimensional representations of algebras over a countable field. It turns out that the limit points are infinite dimensional representations and together with the finite dimensional representations they form a compact metric space. I will also talk about the notion of hyperfiniteness for finite dimensional algebras and its relation with the classical notion of amenability.

Andrew Hubery (Bielefeld)
Euler characteristics of quiver Grassmannians

We show how to define an Euler characteristic for the Grassmannian of submodules of a rigid module, for any finite-dimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is non-empty, generalising the known case of quiver Grassmannians. As an application, this shows that, for any symmetrisable acyclic cluster algebra, the Laurent expansion of a cluster variable always has non-negative coefficients.

Peter Patzt (Berlin)
Representation stability for the general linear groups

The notion of representation stability for the symmetric groups, the general linear groups and the symplectic groups was introduced by Church-Farb. We give a criterion for a sequence of algebraic representations of the general linear groups to be representation stable. With it we prove that the factors of the lower central series of the Torelli subgroups of the automorphism groups of free groups are representation stable.

Pierre-Guy Plamondon (Paris)
Multiplication formulas

In the past decade, the study of cluster algebras via representations of quivers has proved a successful way to tackle some of the problems in the theory. In this talk, I will review the theory of cluster characters and present new multiplication formulas relating them.