Endomorphisms of induced representations

Induction gives an injective algebra homomorphism from the endomorphisms of a finite-dimensional subgroup representation to the endomorphisms of its induction. When all nonidentity Mackey intertwining spaces vanish, this homomorphism is an isomorphism. This identifies the algebra structure over arbitrary fields, including nonsplitting fields.

This is the algebra comparison used in the sufficient direction of Mackey's irreducibility criterion: a division endomorphism algebra remains a division algebra under induction when the other Mackey terms vanish. No semisimplicity hypothesis is needed for the comparison itself.

The construction uses TauCeti.indFDRepMap and the characteristic-free natural-number formula TauCeti.finrank_hom_indFDRep_mackey_erase.

References

• J.-P. Serre, Linear Representations of Finite Groups, §7.4, Proposition 23.
• C. W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, §10.

theorem FDRep.indEndAlgHom_bijective {k G : Type u} [Field k] [Group G] {H : Subgroup G} [Finite G] (A : FDRep k ↥H) (h : ∀ (D : DoubleCoset.Quotient ↑H ↑H), D ≠ DoubleCoset.mk H H 1 → TauCeti.MackeyDisjoint A (Quotient.out D)) : Function.Bijective ⇑A.indEndAlgHom

If all nonidentity double cosets are Mackey disjoint, every endomorphism of the induced representation is induced from a unique endomorphism of the original representation.

noncomputable def FDRep.indEndAlgEquiv {k G : Type u} [Field k] [Group G] {H : Subgroup G} [Finite G] (A : FDRep k ↥H) (h : ∀ (D : DoubleCoset.Quotient ↑H ↑H), D ≠ DoubleCoset.mk H H 1 → TauCeti.MackeyDisjoint A (Quotient.out D)) : CategoryTheory.End A ≃ₐ[k] CategoryTheory.End (TauCeti.indFDRep A)

With no nonidentity Mackey intertwiners, induction identifies the actual endomorphism algebras, over any coefficient field.

@[simp]

theorem FDRep.indEndAlgEquiv_asHom {k G : Type u} [Field k] [Group G] {H : Subgroup G} [Finite G] (A : FDRep k ↥H) (h : ∀ (D : DoubleCoset.Quotient ↑H ↑H), D ≠ DoubleCoset.mk H H 1 → TauCeti.MackeyDisjoint A (Quotient.out D)) (f : CategoryTheory.End A) : ((A.indEndAlgEquiv h) f).asHom = TauCeti.indFDRepMap f.asHom

The endomorphism algebra comparison acts by induction on underlying intertwiners.

@[simp]

theorem FDRep.indFDRepMap_indEndAlgEquiv_symm_asHom {k G : Type u} [Field k] [Group G] {H : Subgroup G} [Finite G] (A : FDRep k ↥H) (h : ∀ (D : DoubleCoset.Quotient ↑H ↑H), D ≠ DoubleCoset.mk H H 1 → TauCeti.MackeyDisjoint A (Quotient.out D)) (f : CategoryTheory.End (TauCeti.indFDRep A)) : TauCeti.indFDRepMap ((A.indEndAlgEquiv h).symm f).asHom = f.asHom

The inverse algebra comparison recovers the unique subgroup endomorphism inducing a given endomorphism of the induced representation.