Fixed points and grade of Hilbert polynomial of invariant rings

Abstract

Let $k$ be a field and let $V$ be a $k$-vector space of dimension $d$. Let $G\subseteq GL(V)$ be a finite group. Let $r={\dim }_k(V^{\ast }{)}^G$. Assume $r\ge 1$. Let $R=k[V{]}^G$ be the ring of invariants of $G$. Let $H_R(n)=a_{d-1}(n)n^{d-1}+\cdots a_1(n)n+a_0(n)$ be the Hilbert polynomial of $R$ where $a_i(-)$ are periodic functions. We show $a_{d-1}(-),\dots ,a_{d-r}(-)$ are constants. In the terminology of Erhart, $\text{grade}{H}_R\le d-r-1$. We also give an example which shows that our result is sharp.

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