proposition: Lemma irr_repr_lin_char (i : Iirr G) x : x \in G -> 'chi_i \is a linear_char -> irr_repr (socle_of_Iirr i) x = ('chi_i x)%:M.
proof: Proof. move=> Gx CFi; rewrite -irrRepr cfunE Gx. move: (_ x); rewrite -[irr_degree _](@natrK algC) -irr1_degree lin_char1 //. by rewrite (natrK 1) => A; rewrite trace_mx11 -mx11_scalar. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype choice ssrnat seq) mathcomp(path div fintype tuple finfun bigop prime order) mathcomp(ssralg poly finset gproduct fingroup morphism) mathcomp(perm automorphism quotient finalg action zmodp) mathcomp(commutator cyclic center pgroup nilpotent sylow) mathcomp(abelian matrix mxalgebra mxpoly mxrepresentation) mathcomp(vector ssrnum algC classfun archimedean)