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.

(* Specifications: *)
Lemma irr_cyclic_lin i : cyclic G -> 'chi[G]_i \is a linear_char.
Variables (A : {group aT}) (G : {group gT}).
(* This is Isaacs, Lemma (2.15) (d). *)
Lemma char_inv (chi : 'CF(G)) x : chi \is a character -> chi x^-1%g = (chi x)^*.
Lemma irr_inv i x : 'chi[G]_i x^-1%g = ('chi_i x)^*.
(* 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.