Realizing Saturated Fusion Systems Athar Ahmad Warraich University - PowerPoint PPT Presentation

Realizing Saturated Fusion Systems Athar Ahmad Warraich University of Birmingham August 12, 2017 Groups St Andrews 2017 Athar Ahmad Warraich 1

Overview Groups St Andrews 2017 Athar Ahmad Warraich 2

Overview Definitions Groups St Andrews 2017 Athar Ahmad Warraich 2

Overview Definitions ’Realizing’ Fusion Systems Groups St Andrews 2017 Athar Ahmad Warraich 2

Overview Definitions ’Realizing’ Fusion Systems Construction Groups St Andrews 2017 Athar Ahmad Warraich 2

Overview Definitions ’Realizing’ Fusion Systems Construction Exoticity Index Groups St Andrews 2017 Athar Ahmad Warraich 2

Overview Definitions ’Realizing’ Fusion Systems Construction Exoticity Index Examples Groups St Andrews 2017 Athar Ahmad Warraich 2

Fusion Systems Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G . Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Composition of morphisms is the composition of group homomorphisms. Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Composition of morphisms is the composition of group homomorphisms. A saturated fusion system F over a finite p -group T is a fusion system which satisfies additional properties. Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Composition of morphisms is the composition of group homomorphisms. A saturated fusion system F over a finite p -group T is a fusion system which satisfies additional properties. There exists a unique largest fusion system, the ”universal” fusion system U ( T ), where, for every P , Q ≤ T , Hom U ( T ) ( P , Q ) = Inj( P , Q ). Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Composition of morphisms is the composition of group homomorphisms. A saturated fusion system F over a finite p -group T is a fusion system which satisfies additional properties. There exists a unique largest fusion system, the ”universal” fusion system U ( T ), where, for every P , Q ≤ T , Hom U ( T ) ( P , Q ) = Inj( P , Q ). We have a unique smallest fusion system F T ( T ), where, for every P , Q ≤ T , Hom F T ( T ) ( P , Q ) = Hom T ( P , Q ). Groups St Andrews 2017 Athar Ahmad Warraich 3

Fusion Systems Let G be a finite group and T a p -subgroup of G .A fusion category F T ( G ) is a category whose objects are subgroups of T and whose morphisms are as follows Hom G ( P , Q ) = { φ ∈ Hom( P , Q ) | φ = c g | P , Q where g ∈ G and P g ≤ Q } . where c g | P , Q : P → Q , u �→ g − 1 ug . Idea of an abstract fusion system: Forget about G , while keeping the maps. A fusion system F = F ( T ) over a finite p -group T is a category whose objects are subgroups of T and whose morphisms are injective group homomorphisms such that for any P , Q ≤ T : Hom F ( P , Q ) ⊇ Hom T ( P , Q ), Every morphism is a composition of an isomorphism and an inclusion map, both of which are in F . Composition of morphisms is the composition of group homomorphisms. A saturated fusion system F over a finite p -group T is a fusion system which satisfies additional properties. There exists a unique largest fusion system, the ”universal” fusion system U ( T ), where, for every P , Q ≤ T , Hom U ( T ) ( P , Q ) = Inj( P , Q ). We have a unique smallest fusion system F T ( T ), where, for every P , Q ≤ T , Hom F T ( T ) ( P , Q ) = Hom T ( P , Q ). F T ( T ) ≤ F ( T ) ≤ U ( T ). Groups St Andrews 2017 Athar Ahmad Warraich 3

Finite Groups ’Realizing’ Fusion Systems Groups St Andrews 2017 Athar Ahmad Warraich 4

Finite Groups ’Realizing’ Fusion Systems Lemma Let T be a p-subgroup of a finite group G. Then F T ( G ) is a fusion system. If T ∈ Syl p ( G ) , then F T ( G ) is a saturated fusion system. Groups St Andrews 2017 Athar Ahmad Warraich 4