Section IrrClassDef.
Variables (gT : finGroupType) (G : {group gT}).
Let sG := DecSocleType (regular_repr algCF G).
Let sG := DecSocleType (regular_repr algCF G).
Let C'G := algC'G G.
Let closG := @groupC _ G.
Local Notation W i := (@socle_of_Iirr _ G i).
Local Notation "''n_' i" := 'n_(W i).
Local Notation "''R_' i" := 'R_(W i).
Local Notation "''e_' i" := 'e_(W i).
Let aG := regular_repr algCF G.
Let R_G := group_ring algCF G.

(* This is Isaacs, Theorem (2.12). *)
Lemma Wedderburn_id_expansion i :
'e_i = #|G|%:R^-1 *: \sum_(x in G) 'chi_i 1%g * 'chi_i x^-1%g *: aG x.