Leicester Pure Mathematics Colloquium, 22/11/2007

5:00pm, MAB 119

Larry Smith (Goettingen/Sheffield):

Abstract:

Let rho: G --> GL(n, F) be a faithful representation of a finite group G over the field F. By means of rho the group G acts on the algebra F[V] of polynomial functions on the representation space V = F^n, fixing the subalgebra F[V]^G (the invariant algebra) and hence also on the algebra F[V]_G = F tensor_{F[V]^G} F[V] of coinvariants. Chevalley's Theorem states that, in the case that F=R and G is a real reflection group, the coinvariant algebra as a G-representation is isomorphic to the regular representation of G.

In this talk I will describe joint work with Abraham Broer of the Universite de Montreal, Victor Reiner and Peter Webb of the University of Minnesota, in which we generalize Chevalley's Theorem in three ways.

(1) we need no assumption on the ground field F, in particular it can be of any characteristic, even one dividing the order of G,

(2) we need no assumption on the representation rho, so in particular G need not be a reflection group, nor must F[V]^G be a polynomial algebra, and finally

(3) we can deal with a relative situation of H < G being a subgroup of G and dealing with the algebra F \tensor_{F[V]^G} F[V]^H as a representation of the group W_G(H) = N_G (H) /H where N_G (H) is the normalizer of G in H.

Our methods are a mixture of homological algebra, representation theory, and invariant theory.