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

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--Römer 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.

8 pages, no figures