Section Basics.
Variables (gT : finGroupType) (B : {set gT}) (S : seq 'CF(B)) (A : {set gT}).
Definition Zchar : {pred 'CF(B)} :=
[pred phi in 'CF(B, A) | dec_Cint_span (in_tuple S) phi].
Fact Zchar_key : pred_key Zchar. Proof. by []. Qed.
Canonical Zchar_keyed := KeyedPred Zchar_key.
Fact Zchar_zmod : zmod_closed Zchar.
Canonical Zchar_opprPred := OpprPred Zchar_zmod.
Canonical Zchar_addrPred := AddrPred Zchar_zmod.
Canonical Zchar_zmodPred := ZmodPred Zchar_zmod.

Lemma zchar_tuple_expansion n (S : n.-tuple 'CF(G)) A phi :
phi \in 'Z[S, A] ->
{z | forall i, z i \in Cint & phi = \sum_(i < n) z i *: S`_i}.