Fixed points and grade of Hilbert polynomial of invariant rings

TL;DR

The paper analyzes Hilbert quasi-polynomials of invariant rings $R=k[V]^G$ for a finite group action, focusing on the constancy of leading coefficients. By establishing a key lemma on dimension drops under fixing $G$-invariant linear forms and a quasi-polynomial refinement, it proves that the top $r$ coefficients $a_{d-1}(-),\ldots,a_{d-r}(-)$ are constant, where $r=\dim_k (V^*)^G$, yielding the Erhart-grade bound $\operatorname{grade} H_R \le d-r-1$. The authors also present a sharp example to show the bound cannot be improved. The results deepen the link between invariant theory and Ehrhart-type quasi-polynomials, clarifying how the fixed-point structure in $V^*$ governs the asymptotic growth of invariants.

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 $r = \dim_k (V^*)^G$. Assume $r \geq 1$. 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 $a_{d-1}(-), \ldots, a_{d-r}(-)$ are constants. In the terminology of Erhart, $\text{grade} H_R \leq d - r-1$. We also give an example which shows that our result is sharp.

Theorems & Definitions (9)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Theorem 3.1
• proof
• proof : Proof of Theorem \ref{['main']}
• Example 5.1