proposition: Lemma mxtrace_regular : {in G, forall x, \tr (aG x) = \sum_i \tr (socle_repr i x) *+ 'n_i}.
proof: Proof. move=> x Gx; have soc1: (Socle sG :=: 1%:M)%MS by rewrite -irr_mx_sum. rewrite -(mxtrace_submod1 (Socle_module sG) soc1) // mxtrace_Socle //. by apply: eq_bigr => i _; rewrite irr_mx_mult. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div choice fintype tuple finfun bigop prime) mathcomp(ssralg poly polydiv finset fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup matrix mxalgebra) mathcomp(mxpoly)