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.

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.