Let xiMV x : x \in G -> xi x * xi (x^-1)%g = 1.

(* Specifications: *)
Lemma lin_charV x : x \in G -> xi x^-1%g = (xi x)^-1.
Lemma lin_char_unity_root x : x \in G -> xi x ^+ #[x] = 1.
Lemma lin_charV_conj x : x \in G -> xi x^-1%g = (xi x)^*.
Lemma lin_char_irr : xi \in irr G.
Lemma mul_conjC_lin_char : xi * xi^*%CF = 1.
Lemma lin_char_unitr : xi \in GRing.unit.
End OneChar.
Lemma cfAut_lin_char u (xi : 'CF(G)) :
(cfAut u xi \is a linear_char) = (xi \is a linear_char).
Lemma irr_repr_lin_char (i : Iirr G) x :
x \in G -> 'chi_i \is a linear_char ->
irr_repr (socle_of_Iirr i) x = ('chi_i x)%:M.