An algorithm for determining the simplicity of a modular group representation (Q1109882)

Let G be a finite group and \(\kappa\) : \(G\to GL_ n(F)\) a faithful representation of G over a splitting field F, so that the space \(F^ n\) becomes a module over the group algebra FG. Identify each \(g\in G\) with \(\kappa\) (g). Let \(t\leq n\) and let \(N_ t\) be the intersection of the nullspaces of all \(S_{2t}[g_ 1,g_ 2,...,g_{2t}]\), where each \(g_ i\in G\), and \(S_{2t}[X_ 1,X_ 2,...,X_{2t}]\) is the standard polynomial of degree 2t, in the sense of Amitsur and Levitski. Then the socle of \(N_ t\) is the sum of all those simple FG-submodules of \(F^ n\) that have F-dimension at most t. Consequently, \(F^ n\) is FG-simple if and only if \(N_{n-1}=0\).