Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

TL;DR

This paper generalizes classical invariant theory theorems for finite reflection groups to apply to all finite groups across any characteristic, broadening their scope significantly.

Contribution

It extends key invariant theory results of Chevalley, Shephard--Todd, Mitchell, and Springer to arbitrary finite groups and characteristics.

Findings

Broadened the applicability of classical theorems to all finite groups.

Established new isomorphisms between group algebras and coinvariant modules.

Provided a unified framework for invariant theory across different group types.

Abstract

We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.