Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay

Abstract:

Let $\rho : G \hookrightarrow \operatorname {GL}(n, \mathbb {F})$ be a faithful representation of a finite group $G$ and $\chi : G \longrightarrow \mathbb {F}^\times$ a linear character. We study the module $\mathbb {F}[V]^G_\chi$ of $\chi$-relative invariants. We prove a modular analogue of result of R. P. Stanley and V. Reiner in the case of nonmodular reflection groups to the effect that these modules are free on a single generator over the ring of invariants $\mathbb {F}[V]^G$. This result is then applied to show that the ring of invariants for $H = \operatorname {ker}(\chi ) \leq G$ is Cohen–Macaulay. Since the Cohen–Macaulay property is not an issue in the nonmodular case (it is a consequence of a theorem of Eagon and Hochster), this would seem to be a new way to verify the Cohen–Macaulay property for modular rings of invariants. It is known that the Cohen–Macaulay property is inherited when passing from the ring of invariants of $G$ to that of a pointwise stabilizer $G_U$ of a subspace $U \leq V = \mathbb {F}^n$. In a similar vein, we introduce for a subspace $U \leq V$ the subgroup $G_{\langle U \rangle }$ of elements of $G$ having $U$ as an eigenspace, and prove that $\mathbb {F}[V]^G$ Cohen–Macaulay implies $\mathbb {F}[V]^{G_{\langle U \rangle }}$ is also.