Symmetric group fixed quotients of polynomial rings

TL;DR

This paper investigates the structure of cofixed spaces under symmetric group actions on polynomial rings, revealing stability and complexity jumps related to the characteristic of the base ring and the size of the group.

Contribution

It characterizes the cofixed space as an ideal of symmetric polynomials over various base rings and uncovers stability and complexity jumps as the group size varies.

Findings

Cofixed space is isomorphic to an ideal of symmetric polynomials in characteristic zero.

Ideals exhibit stability as the number of variables grows.

Ideals' complexity jumps at multiples of the prime characteristic.

Abstract

Given a representation of a finite group $G$ over some commutative base ring $\mathbf{k}$, the cofixed space is the largest quotient of the representation on which the group acts trivially. If $G$ acts by $\mathbf{k}$-algebra automorphisms, then the cofixed space is a module over the ring of $G$-invariants. When the order of $G$ is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that $G$ is the symmetric group on $n$ letters acting on a polynomial ring by permuting its variables. When $\mathbf{k}$ has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials. Localizing $\mathbf{k}$ at a prime integer $p$ while letting $n$ vary reveals striking behavior in these ideals. As $n$ grows, the ideals stay stable in a sense, then jump in complexity each time $n$ reaches a multiple of $p$.