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

Larry Smith
2006 Proceedings of the American Mathematical Society  
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
more » ... lay 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 ≤ V = F n . In a similar vein, we introduce for a subspace U ≤ V the subgroup G U of elements of G having U as an eigenspace, and prove that F[V ] G Cohen-Macaulay implies F[V ] G U is also.
doi:10.1090/s0002-9939-06-08427-9 fatcat:xcyh7pfy7rebjlu2jrlb3yrpaa