Section AutChar.
Variables (gT : finGroupType) (G : {group gT}).
Implicit Type u : {rmorphism algC -> algC}.
Implicit Type chi : 'CF(G).
Let xiMV x : x \in G -> xi x * xi (x^-1)%g = 1.
End OneChar.
Fact linear_char_key B : pred_key (@linear_char B). Proof. by []. Qed.
Canonical linear_char_keted B := KeyedQualifier (linear_char_key B).
Fact linear_char_divr : divr_closed (@linear_char G).
Canonical lin_char_mulrPred := MulrPred linear_char_divr.
Canonical lin_char_divrPred := DivrPred linear_char_divr.
Variables (A : {group aT}) (G : {group gT}).
(* The character table. *)
Definition irr_class i := enum_val (cast_ord (NirrE G) i).
Definition class_Iirr xG :=
cast_ord (esym (NirrE G)) (enum_rank_in (classes1 G) xG).
Local Notation c := irr_class.
Local Notation g i := (repr (c i)).
Local Notation iC := class_Iirr.
Definition character_table := \matrix_(i, j) 'chi[G]_i (g j).
Local Notation X := character_table.

Lemma irr_classP i : c i \in classes G.