End InertiaBig.
End InertiaBigdprod.
Section ConsttInertiaBijection.
Variables (gT : finGroupType) (H G : {group gT}) (t : Iirr H).
Hypothesis nsHG : H <| G.
Local Notation theta := 'chi_t.
Local Notation T := 'I_G[theta]%G.
Local Notation "` 'T'" := 'I_(gval G)[theta]
(at level 0, format "` 'T'") : group_scope.
Let calA := irr_constt ('Ind[T] theta).
Let calB := irr_constt ('Ind[G] theta).
Local Notation AtoB := (Ind_Iirr G).

(* Specifications: *)
(* This is Isaacs, Theorem (6.11). *)
Theorem constt_Inertia_bijection :
[/\ (*a*) {in calA, forall s, 'Ind[G] 'chi_s \in irr G},
(*b*) {in calA &, injective (Ind_Iirr G)}, Ind_Iirr G @: calA =i calB, (*c*) {in calA, forall s (psi := 'chi_s) (chi := 'Ind[G] psi), [predI irr_constt ('Res chi) & calA] =i pred1 s} & (*d*) {in calA, forall s (psi := 'chi_s) (chi := 'Ind[G] psi), '['Res psi, theta] = '['Res chi, theta]}].  End ConsttInertiaBijection.  Section ExtendInvariantIrr.  Variable gT : finGroupType. Implicit Types G H K L M N : {group gT}.  Section ConsttIndExtendible.  Variables (G N : {group gT}) (t : Iirr N) (c : Iirr G). Let theta := 'chi_t. Let chi := 'chi_c.  Definition mul_Iirr b := cfIirr ('chi_b * chi). Definition mul_mod_Iirr (b : Iirr (G / N)) := mul_Iirr (mod_Iirr b).  Hypotheses (nsNG : N <| G) (cNt : 'Res[N] chi = theta). Let sNG : N \subset G. Proof. exact: normal_sub. Qed. Let nNG : G \subset 'N(N). Proof. exact: normal_norm. Qed.  Lemma extendible_irr_invariant : G \subset 'I[theta]. Let IGtheta := extendible_irr_invariant.  (* This is Isaacs, Theorem (6.16) *)
Theorem constt_Ind_mul_ext f (phi := 'chi_f) (psi := phi * theta) :
G \subset 'I[phi] -> psi \in irr N ->
let calS := irr_constt ('Ind phi) in
[/\ {in calS, forall b, 'chi_b * chi \in irr G},
{in calS &, injective mul_Iirr},
irr_constt ('Ind psi) =i [seq mul_Iirr b | b in calS]
& 'Ind psi = \sum_(b in calS) '['Ind phi, 'chi_b] *: 'chi_(mul_Iirr b)].
(* This is Isaacs, Corollary (6.17) (due to Gallagher). *)
Corollary constt_Ind_ext :
[/\ forall b : Iirr (G / N), 'chi_(mod_Iirr b) * chi \in irr G,
injective mul_mod_Iirr,
irr_constt ('Ind theta) =i codom mul_mod_Iirr
& 'Ind theta = \sum_b 'chi_b 1%g *: 'chi_(mul_mod_Iirr b)].