The Cohen-Macaulay property of separating invariants of finite groups

TL;DR

This paper investigates the Cohen-Macaulay property of separating invariants of finite groups, revealing conditions under which such algebras are or are not Cohen-Macaulay, especially in modular representations and p-groups.

Contribution

It establishes new criteria linking the Cohen-Macaulay property of separating algebras to group structure and representation characteristics, including the role of bireflections.

Findings

No graded separating algebra is Cohen-Macaulay over fields of positive characteristic with many copies of a faithful modular representation.

For p-groups, Cohen-Macaulay separating algebras imply the group is generated by bireflections.

Examples show Cohen-Macaulay separating algebras can exist even when the invariant ring is not Cohen-Macaulay.

Abstract

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.