Hilbert series of representations of categories of $G$-sets

TL;DR

This paper proves that the Hilbert series associated with functors from the category of finite free G-sets to vector spaces is rational, with denominators linked to the character table of the finite group G.

Contribution

It establishes the rationality of the Hilbert series for these functors and describes the form of their denominators in terms of G's character table.

Findings

Hilbert series are rational functions.

Denominators are linear polynomials with coefficients from the character field.

Results connect G-set categories with representation theory.

Abstract

Let $G$ be a finite group. A contravariant functor from the category of finite free $G$-sets to vector spaces has an associated Hilbert series, which records the underlying sequence of $G^n$ representations, $n \in \mathbb N$. We prove that this Hilbert series is rational with denominator given by linear polynomials with coefficients in the field generated by the character table of $G$.