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}).

(* This is Isaacs, Corollary (2.14). *)
Corollary first_orthogonality_relation (i j : Iirr G) :
#|G|%:R^-1 * (\sum_(x in G) 'chi_i x * 'chi_j x^-1%g) = (i == j)%:R.