Lemma xcfun_rE A chi : xcfun_r A chi = xcfun chi A=> phi psi; rewrite /= /xcfun !mxE -sumrB; apply: eq_bigr => i _.
by rewrite !mxE !cfunE mulrBr.Definition socle_of_Iirr (i : Iirr G) : sG :=
  enum_val (cast_ord Iirr_cast (i + offset)).

(* Specifications: *)
Proof. by rewrite /W add0r cast_ordKV enum_ra
Proof. by move=> i; rewrite /irr_of_socle en