proposition: Lemma cfDetMorph aT rT (D G : {group aT}) (f : {morphism D >-> rT}) (phi : 'CF(f @* G)) : phi \is a character -> cfDet (cfMorph phi) = cfMorph (cfDet phi).
proof: Proof. move=> Nphi; have [sGD | not_sGD] := boolP (G \subset D); last first. have /natrP[n Dphi1] := Cnat_char1 Nphi. rewrite !cfMorphEout // Dphi1 lin_char1 ?cfDet_lin_char // scale1r. by rewrite scaler_nat cfDetMn ?cfDet_id ?rpred1 // expr1n. have [rG ->] := char_reprP Nphi; rewrite !(=^~ cfRepr_morphim, cfDetRepr) //. apply: cfRepr_sim; exists 1%:M; rewrite ?row_free_unit ?unitmx1 // => x Hx. by rewrite mulmx1 mul1mx. 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)