Notation "''Chi_' i" := (irr_repr (socle_of_Iirr i))
  (at level 8, i attuple 'CF(B) :=
   let irr_of i := 'Res[B, <<B>>] (@cfRepr gTite size_tuple => lt_i_G <-.
by exists (Ordinal lt_i_G); rewrite (tnth_nth 0).
Qed.ite sum_cfunE (reindex _ (socle_of_Iirr_bij _)); apply: eq_bigr => i _.
by rewrite -irrRepr cfRepr1 !cfunE Gx mulr_natl.
Qed.

Proof. by apply/injectiveP/free_uniq; rewrite m