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arXiv · 1606.07956

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

Robert Krone·Anton Leykin·Andrew Snowden
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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.

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Robert Krone, Anton Leykin, Andrew Snowden. 2016-06-25. Hilbert series of symmetric ideals in infinite polynomial rings via formal languages. https://arxiv.org/abs/1606.07956

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