Abstract:
Let G be a finite group, V a finite dimensional G- module over a field F,
and S(V) the symmetric algebra of V. The above problem seeks to determine
when the ring of invariants S(V)^G is a polynomial ring. In the non-modular case
(i.e. char(F) being prime to order of G), this was settled in the Shephard-Todd-Chevalley
theorem. The modular case (i.e. char(F) divides order of G), is still wide open.
I shall discuss some older results due to Serre, Nakajima, Kemper-Malle and
explain some new results, mostly in dimension 3.