proposition: Lemma cfDet_mul_lin gT (G : {group gT}) (lambda phi : 'CF(G)) : lambda \is a linear_char -> phi \is a character -> cfDet (lambda * phi) = lambda ^+ Num.trunc (phi 1%g) * cfDet phi.
proof: Proof. case/andP=> /char_reprP[[n1 rG1] ->] /= n1_1 /char_reprP[[n2 rG2] ->] /=. do [rewrite !cfRepr1 pnatr_eq1 natrK; move/eqP] in n1_1 *. rewrite {n1}n1_1 in rG1 *; rewrite cfRepr_prod cfDetRepr. apply/cfun_inP=> x Gx; rewrite !cfunE cfDetRepr cfunE Gx !mulrb !trace_mx11. rewrite !mxE prod_repr_lin ?mulrb //=; case: _ / (esym _); rewrite detZ. congr (_ * _); case: {rG2}n2 => [|n2]; first by rewrite cfun1E Gx. by rewrite expS_cfunE //= cfunE Gx trace_mx11. 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)