Hilbert series of symmetric ideals in infinite polynomial rings via formal languages

TL;DR

This paper provides a concise proof that the Hilbert series of symmetric ideals in infinite polynomial rings is rational, using formal language theory and an algorithm for computation.

Contribution

It introduces a new, simplified proof of the rationality of the Hilbert series for symmetric ideals, leveraging formal languages and an efficient algorithm.

Findings

Hilbert series of symmetric ideals is rational

A new proof simplifies previous arguments

An algorithm for computing the series is presented

Abstract

Let $R$ be the polynomial ring $K[x_{i,j}]$ where $1 \le i \le r$ and $j \in \mathbb{N}$, and let $I$ be an ideal of $R$ stable under the natural action of the infinite symmetric group $S_{\infty}$. Nagel--R\"omer recently defined a Hilbert series $H_I(s,t)$ of $I$ and proved that it is rational. We give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series.