Invariants of modular groups

Let V be a vector space of dimension n over a field F of characteristic p 0 and let F[V ] be the symmetric algebra of the dual V of V . Let G GL(V ) be a finite group acting on F[V ]. In this paper the author proves that certain quotient rings of the invariant ring F[V ]G are polynomial, following H. Nakajima’s paper [J. Algebra 85 (1983), no. 2, 253–286; MR0725082 (85f:20038)]. The result he obtains is used to give a simpler proof of Landweber-Stong’s result: If dim V G = n−1, then F[V ]G is polynomial. Derksen and Kemper’s conjecture states that the Hilbert ideal of G, F[V ]G+ ·F[V ], is generated by homogeneous elements of degree at most the order of G. Here the author shows that this conjecture holds if P g2G(g −1)V (V )G. In particular, it holds if G is an abelian p-group generated by reflections. He also gives a characterization of polynomial invariant rings F[V ]N, where N is a maximal subgroup of a p-group G such that F[V ]G is polynomial. Finally, he proves some results about the Cohen-Macaulay properties of invariant rings and about the invariant rings of affine groups.