On an application of Poincaré series (Q1951358)

This article investigates the rings of coinvariants of modular representations of finite groups. Let \(G\) ge a finite group, and let \(V\) be a vector space of dimension \(n\) over a field \(\mathbb F\) and \(\mathbb F[V]\) denote the symmetric algebra of the dual space \(V^*\) of \(V\). Let \(G\hookrightarrow \mathrm{GL}(V)\) be a faithful representation of \(G\) over \(\mathbb F\). The subalgebra \(\mathbb F[V]^G\) fixed under the \(G\)-action is called the ring of invariants of \(G\). Denote by \(h(G)\) the ideal in \(\mathbb F[V]\) generated by all \(G\)-invariant homogeneous polynomials of positive degree. The graded quotient algebra \(\mathbb F[V]_G=\mathbb F[V]/h(G)\) is called the ring of coinvariants. In the nonmodular case, i.e., the order of \(G\) is prime to the characteristic of \(\mathbb F\), \textit{C. Chevalley} [Am. J. Math. 77, 778--782 (1955; Zbl 0065.26103)] proved that if \(\mathbb F[V]^G\) is a polynomial algebra, then \(\mathbb F[V]_G\) is isomorphic to the regular representation \(\mathrm{Reg}_{\mathbb F}(G)\). In the paper under review the author proves that in the modular case the isomorphism class \([\mathbb F[V]_G]\) is a multiple of \([\mathrm{Reg}_{\mathbb F}(G)]\) in the Grothendieck group \(R_{\mathbb F}(G)\), provided that \(h(G)\) is generated by all primary invariants of \(\mathbb F[V]^G\).