Transvection free groups and invariants of polynomial tensor exterior algebras

Abstract

Let ρ:G↪Gl(n,$\mathbb{F}$) be a representation of a finite groupG over a field$\mathbb{F}$ such that the ring of invariants$\mathbb{F}\left[ V \right]^G$ is a polynomial algebra$\mathbb{F}\left[ {f_1 ,... ,f_n } \right]$. It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of$\mathbb{F}$), the invariants ofG acting on the tensor product$\mathbb{F}\left[ V \right] \otimes E\left[ V \right]$ of a polynomial and an exterior algebra are given by$\mathbb{F}\left[ {f_1 ,... ,f_n } \right] \otimes E\left[ {df_1 ,... ,df_n } \right]$,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in$\mathbb{F}\left[ V \right] \otimes E\left[ V \right]$ is of this form if and only if$\mathbb{F}\left[ V \right]^G$ is a polynomial algebra and all pseudoreflections in ϱ(G) are diagonalizable.