When are permutation invariants Cohen-Macaulay?

TL;DR

This paper investigates when permutation invariants over finite fields are Cohen-Macaulay, providing an efficient algorithm to determine such primes and generalizing classical discriminants for certain reflection groups.

Contribution

It introduces an algorithm to identify primes where permutation invariant rings are Cohen-Macaulay over finite fields and generalizes classical discriminants for specific reflection groups.

Findings

Algorithm efficiently determines primes for Cohen-Macaulay invariants over finite fields

Generalization of classical discriminant for subgroups of complex reflection groups

Provides new insights into invariant theory in positive characteristic

Abstract

Over a field of characteristic 0, every ring of invariants of a finite group is Cohen-Macaulay. This is not true for fields of positive characteristic. We consider permutation representations and their invariant rings over fields $\mathbb{F}_p$ of prime order. We give an efficient algorithm which for any given permutation representation, determines those primes $p$ for which the invariant ring over $\mathbb{F}_p$ is Cohen-Macaulay, using linear algebra over $\ZZ$. A generalization of the classical discriminant associated to the alternating group is defined for subgroups of certain finite unitary complex reflection groups.