Saturated fusion systems with parabolic families Silvia Onofrei The - - PowerPoint PPT Presentation

Saturated fusion systems with parabolic families

Silvia Onofrei

The Ohio State University AMS Fall Central Sectional Meeting, Saint Louis, Missouri, 19-21 October 2013

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Basics on Fusion Systems A fusion system F over a finite p-group S is a category whose:

• objects are the subgroups of S,
• morphisms are such that HomS(P,Q) ⊆ HomF (P,Q) ⊆ Inj(P,Q),

every F -morphism factors as an F -isomorphism followed by an inclusion. Let F be a fusion system over a finite p-group S. A subgroup P of S is fully F -normalized if |NS(P)| ≥ |NS(ϕ(P))|, for all ϕ ∈ HomF (P,S);

F -centric

if CS(ϕ(P)) = Z(ϕ(P)) for all ϕ ∈ HomF (P,S);

F -essential

if Q is F -centric and Sp(OutF (P)) = Sp(AutF (P)/AutP(P)) is disconnected. The fusion system F over a finite p-group S is saturated if the following hold:

• Sylow Axiom
• Extension Axiom

The normalizer of P in F is the fusion system NF (P) on NS(P)

ϕ ∈ HomNF (Q,R) if ∃ ϕ ∈ HomF (PQ,PR) with ϕ(P) = P and ϕ|Q = ϕ.

The fusion system F is constrained if F = NF (Q) for some F -centric subgroup Q = 1 of S. The group G has (finite) Sylow p-subgroup S if S is a finite p-subgroup of G and if every finite p-subgroup of G is conjugate to a subgroup of S.

FS(G) is the fusion system on S with HomF (P,Q) = HomG(P,Q), for P,Q ≤ S.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 1/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Basics on Chamber Systems

1

A chamber system over a set I is a nonempty set C whose elements are called chambers together with a family of equivalence relations (∼i;i ∈ I) on C indexed by I.

2

The i-panels are the equivalence classes with respect to ∼i.

3

Two distinct chambers c and d are called i- adjacent if they are contained in the same i-panel: c ∼i d

4

A gallery of length n connecting two chambers c0 and cn is a sequence of chambers c0 ∼i1 c1 ∼i2 ... ∼in−1 cn−1 ∼in cn

5

The chamber system C is connected if any two chambers can be joined by a gallery.

6

The rank of the chamber system is the cardinality of the set I.

7

A morphism ϕ : C → D between two chamber systems over I is a map on chambers that preserves i-adjacency: if c,d ∈ C and c ∼i d then ϕ(c) ∼i ϕ(d) in D.

8

Aut(C) is the group of all automorphisms of C (automorphism has the obvious meaning).

9

If G is a group of automorphisms of C then orbit chamber system C/G is a chamber system over I.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 2/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Fusion Systems with Parabolic Families A fusion system F over a finite p-group S has a family {Fi;i ∈ I} of parabolic subsystems if: (F0)

∀i ∈ I, Fi is saturated, constrained, of Fi-essential rank one;

(F1)

B := NF (S) is a proper subsystem of Fi for all i ∈ I;

(F2)

F = Fi;i ∈ I and no proper subset {Fj;j ∈ J ⊂ I} generates F ;

(F3)

Fi ∩Fj = B for any pair of distinct elements Fi and Fj;

(F4)

Fij := Fi,Fj is saturated constrained subsystem of F for all i,j ∈ I.

Proposition (Onofrei, 2011)

If F contains a family of parabolic subsystems then there are:

• p′-reduced p-constrained finite groups B,Gi,Gij with B = FS(B), Fi = FS(Gi), Fij = FS(Gij), ∀i,j ∈ I;
• injective homomorphisms ψi : B → Gi, ψij : Gi → Gij such that ψji ◦ ψj = ψij ◦ ψi, ∀i,j ∈ I.

In other words, A = {(B,Gi,Gij),(ψi,ψij); i,j ∈ I} is a diagram of groups. The proof is based on:

• [BCGLO, 2005]: Every saturated constrained fusion system F over S is the fusion system FS(G)
• f a finite group G that is p′-reduced Op′(G) = 1 and p-constrained CG(Op(G)) ≤ Op(G),

and if we set U := Op(F ) then 1 −

→ Z(U) − → G − → AutF (U) − → 1.

• [Aschbacher, 2008]: If G1 and G2 are such that F = FS(G1) = FS(G2) then there is an isomorphism

ϕ : G1 → G2 with ϕ|S = IdS.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 3/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Fusion - Chamber System Pairs

Lemma (Onofrei, 2011)

If G is a faithful completion of the diagram of groups A then: (P1) G := Gi,i ∈ I = Gj,j ∈ J I (P2) Gi ∩ Gj = B for all i = j in I; (P3) B = Gi for all i ∈ I; (P4)

∩g∈GBg = 1.

Hence (G;B,Gi,i ∈ I) is a parabolic system of rank n = |I|. The chamber system C = C(G;B,Gi,i ∈ I) is defined as follows:

• the chambers are cosets gB for g ∈ G;
• two chambers gB and hB are i-adjacent if gGi = hGi where g,h ∈ G.

G acts chamber transitively, faithfully on C by left multiplication.

Definition (Onofrei, 2011)

A fusion - chamber system pair (F ,C) consists of:

⋄ a fusion system F with a family of parabolic subsystems {Fi;i ∈ I}; ⋄ a chamber system C = C(G;B,Gi,i ∈ I) with G a faithful completion of A.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 4/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Main Theorem on Fusion - Chamber System Pairs

Theorem (Onofrei, 2011)

Let (F ,C) be a fusion-chamber system pair. Assume the following hold.

(i). C P is connected for all p-subgroups P of G. (ii). If P is F -centric and if R is a p-subgroup of AutG(P), then (C P/CG(P))R is connected.

Then F = FS(G) is a saturated fusion system over S. Sketch of the Argument Step 1: S is a Sylow p-subgroup of G. Since C P is connected, C P = /

0 and ∃g ∈ G such that gB ∈ C P, thus P ≤ gBg−1 and since

gSg−1 ∈ Sylp(gBg−1), ∃h ∈ G such that hPh−1 ≤ S. Step 2: F is the fusion system given by conjugation in G, this means F = FS(G). Clearly F ⊆ FS(G).

FS(G) ⊆ F follows from the fact that every morphism in FS(G)

is a composite of morphisms ϕ1,...,ϕn with ϕi ∈ FS(Gji ), ji ∈ I.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 5/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Main Theorem: Sketch of the Argument Step 3: Every morphism in F is a composition of restrictions of morphisms between F -centric subgroups. Recall Fi is saturated, constrained and of essential rank one. If Ei is Fi-essential then Ei is F -centric. Alperin-Goldschmidt Theorem: Each morphism ϕi ∈ Fi can be written as a composition of restrictions

• f Fi-automorphisms of S and of automorphisms of fully Fi-normalized Fi-essential subgroups of S.

Hence we may use: [BCGLO, 2005]: It suffices to verify the saturation axioms for the collection of F -centric subgroups only. Step 4: The Sylow Axiom: For all F -centric P that are fully F -normalized, AutS(P) ∈ Sylp (AutF (P)). Proposition [Stancu, 2004]: Assume that

• AutS(S) is a Sylow p-subgroup of AutF (S);
• The Extension Axiom holds for all F -centric subgroups P.

Then if Q is F -centric and fully F -normalized then AutS(Q) is a Sylow p-subgroup of AutF (Q).

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 6/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Main Theorem: Sketch of the Argument Step 5: The Extension Axiom: Let P be F -centric. For any ϕ ∈ HomF (P,S) there is a morphism

ϕ ∈ HomF (Nϕ,S) such that ϕ|P = ϕ.

PCS(P) ≤ Nϕ ≤ NS(P) For P ≤ S we introduce a new chamber system Rep(P,C) as follows: The chambers are the elements of Rep(P,B) := Inn(B)\Inj(P,B);

[α] ∈ Rep(P,B) denotes the class of α ∈ Inj(P,B)

The i-panels are represented by the elements of

Rep(P,B,Gi) := {[γ] ∈ Rep(P,Gi) : γ ∈ Inj(P,Gi) with γ(P) ≤ B}.

Let τK

H denote the inclusion map of the group H into the group K.

Two chambers [α] and [β] are i-adjacent if

• τGi

B ◦α

• =
• τGi

B ◦β

• in Rep(P,Gi).

NG(P) acts on Rep(P,C) via g ·[α] = [α◦cg−1] for g ∈ NG(P).

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 7/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Main Theorem: Sketch of the Argument There is an NG(P)-equivariant chamber system isomorphism fP : C P −

→ Rep(P,C)0

given by f P(gB) = [cg−1] that induces an isomorphism on the orbit chamber systems

C P/CG(P) −

→ Rep(P,C)0

where Rep(P,C)0 is the connected component of Rep(P,C) that contains [τB

P],

affords the action of AutG(P) = NG(P)/CG(P). For ϕ ∈ HomF (P,S) let K = Nϕ/Z(P) = AutNϕ(P). The map

Γ : Rep(Nϕ,C)0 − → Rep(P,C)K

0 ≃

C P/CG(P) K

is onto.

• that is induced by the restriction Nϕ → P
• between the connected component of Rep(Nϕ,C) that contains [τB

Nϕ]

and the fixed point set of K acting on Rep(P,C)0

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 8/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Main Theorem: Sketch of the Argument There is an NG(P)-equivariant chamber system isomorphism fP : C P −

→ Rep(P,C)0

given by f P(gB) = [cg−1] that induces an isomorphism on the orbit chamber systems

C P/CG(P) −

→ Rep(P,C)0

where Rep(P,C)0 is the connected component of Rep(P,C) that contains [τB

P],

affords the action of AutG(P) = NG(P)/CG(P). For ϕ ∈ HomF (P,S) let K = Nϕ/Z(P) = AutNϕ(P). The map

Γ : Rep(Nϕ,C)0 − → Rep(P,C)K

0 ≃

C P/CG(P) K

is onto.

• that is induced by the restriction Nϕ → P
• between the connected component of Rep(Nϕ,C) that contains [τB

Nϕ]

and the fixed point set of K acting on Rep(P,C)0

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 8/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

An Application: Fusion Systems with Classical Parabolic Families Assume F contains a family of parabolic systems. Set Ui = Op(Fi) and Uij = Op(Fij). We say F contains a classical family of parabolic systems with diagram M if:

♦ For each i ∈ I, OutFi (Ui) is a rank one finite group of Lie type in characteristic p. ♦ For each pair i,j ∈ I, OutFij (Uij) is either a rank two finite group of Lie type in characteristic p

• r it is a (central) product of two rank one finite groups of Lie type in characteristic p.

The diagram M is a graph whose vertices are labeled by the elements of I,

• OutFij (Uij) is a product of two rank one Lie groups then the nodes i and j are not connected,
• OutFij (Uij) is a rank two Lie group the nodes i and j are connected by a bond of strength mij − 2,

where mij denotes the integer that defines the Weyl group.

Proposition

Let (F ,C) be a fusion-chamber system pair with |I| ≥ 3. Assume that:

(i). F contains a classical family of parabolic systems with diagram M ; (ii). M is a spherical diagram.

Then F is the fusion system of a finite simple group of Lie type in characteristic p extended by diagonal and field automorphisms.

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 9/10

Silvia Onofrei (OSU), Saturated fusion systems with parabolic families

Thank You The End

Fall Central Sectional Meeting, Saint Louis, Missouri, 19 October 2013 10/10