Hilbert series of group representations and Gröbner bases for generic modules
Let \(G\) be a finite group, \(V\) an \(n\)-dimensional vector space over an algebraically closed field \(K\) whose characteristic does not divide the order of \(G\), \(A\) the \(K\)-algebra of polynomial functions from \(V\) to \(K\), and let \(\pi : G \to GL_ K (V)\) be a representation of \(G\). The set of polynomial maps from \(V\) to \(V\) is a graded \(n\)-dimensional free module over \(A\) and contains \(\pi (G)\). In this paper \(M (\pi)\), the \(A\)- submodule generated by \(\pi (G)\), is studied. First it is shown that the decomposition of \(\pi\) into irreducible representations induces a decomposition of \(M (\pi)\) into generic modules. This implies a formula for the rank and the first Betti number of \(M (\pi)\). Then, applying Gröbner basis methods to generic modules, the author computes the second Betti number and the Hilbert series for \(M (\pi)\).
scientific article; zbMATH DE number 558524
scientific article