Section DetRepr.
Variables (n : nat) (rG : mx_representation [fieldType of algC] G n).
Definition det_repr_mx x : 'M_1 := (\det (rG x))%:M.
Fact det_is_repr : mx_repr G det_repr_mx.
Canonical det_repr := MxRepresentation det_is_repr.
Definition detRepr := cfRepr det_repr.
End DetRepr.
Definition cfDet phi := \prod_i detRepr 'Chi_i ^+ truncC '[phi, 'chi[G]_i].
Definition cfDet_order phi := #[cfDet phi]%CF.
Definition cfDet_order_lin xi :
xi \is a linear_char -> cfDet_order xi = #[xi]%CF.

Lemma cfDet_mul_lin gT (G : {group gT}) (lambda phi : 'CF(G)) :
lambda \is a linear_char -> phi \is a character ->
cfDet (lambda * phi) = lambda ^+ truncC (phi 1%g) * cfDet phi.