Section IrrComponent.
(* The component of the socle of the regular module that is associated to an *)
(* irreducible representation.                                               *)
Variables (n : nat) (rG : mx_representation F G n).
Local Notation E_G := (enveloping_algebra_mx rG).
Let not_rsim_op0 (iG j : sG) A :
mx_rsim rG (socle_repr iG) -> iG != j -> (A \in 'R_j)%MS ->
gring_op rG A = 0.
Definition irr_comp := odflt 1%irr [pick i | gring_op rG 'e_i != 0].
Local Notation iG := irr_comp.
Hypothesis irrG : mx_irreducible rG.
End IrrComponent.
Hypothesis splitG : group_splitting_field G.
Definition linear_irr := [set i | 'n_i == 1%N].

Lemma Wedderburn_center :
('Z(R_G) :=: \matrix_(i < #|sG|) mxvec 'e_(enum_val i))%MS.