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

Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 134 (2006), 2205-2212
MSC (2000): Primary 13A50, 13C14
Published electronically: March 21, 2006
MathSciNet review: 2213692
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Abstract: Let $ \rho : G \hookrightarrow {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.

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Larry Smith
Affiliation: Mathematisches Institut, Bunsenstraße 3–5, D 37073 Göttingen, Federal Republic of Germany

Received by editor(s): March 4, 2005
Published electronically: March 21, 2006
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.