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

2006
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Proceedings of the American Mathematical Society
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Let ρ : G → GL(n, F) be a faithful representation of a finite group G and χ : G −→ F × a linear character. We study the module F[V ] G χ of χrelative 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 F[V ] G . This result is then applied to show that the ring of invariants for H = ker(χ) ≤ G is Cohen-Macaulay. Since the

doi:10.1090/s0002-9939-06-08427-9
fatcat:xcyh7pfy7rebjlu2jrlb3yrpaa