Section GenericClassSums.
(* This is Isaacs, Theorem (2.4), generalized to an arbitrary field, and with *)
(* the combinatorial definition of the coeficients exposed.                   *)
(* This part could move to mxrepresentation.*)
Variable (gT : finGroupType) (G : {group gT}) (F : fieldType).
Definition gring_classM_coef_set (Ki Kj : {set gT}) g :=
[set xy in [predX Ki & Kj] | let: (x, y) := xy in x * y == g]%g.
Definition gring_classM_coef (i j k : 'I_#|classes G|) :=
#|gring_classM_coef_set (enum_val i) (enum_val j) (repr (enum_val k))|.
Definition gring_class_sum (i : 'I_#|classes G|) := gset_mx F G (enum_val i).
Local Notation "''K_' i" := (gring_class_sum i)
(at level 8, i at level 2, format "''K_' i") : ring_scope.
Local Notation a := gring_classM_coef.

Theorem gring_classM_expansion i j : 'K_i *m 'K_j = \sum_k (a i j k)%:R *: 'K_k.
Fact gring_irr_mode_key : unit. Proof. by []. Qed.