# Rings of Invariants and Varieties of Representations

Algebra/Geometry/Topology Seminar

#### by Jim Shank

Institution: Kent
Date: Thu 22nd April 2010
Time: 2:15 PM
Location: Babel Middle Theatrette (106)

Abstract: Suppose that G is a finite group, F is a field and V is finite dimensional representation of G over F. The action of G on V induces an action on the dual V^* which extends to an action by algebra automorphisms on the symmetric algebra S:=S(V^*). The subring of fixed points, S^G, is known as the ring of invariants of V. For fixed G, F, and dim(V), the representations of G can be paramterised by an algebraic variety. I will discuss the resulting parameterisation of invariant rings, using modular representations of elementary abelian p-groups as illustrative examples.