Subgroup Complexes and their Lefschetz Modules Silvia Onofrei - - PowerPoint PPT Presentation

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Subgroup Complexes and their Lefschetz Modules

Silvia Onofrei

Department of Mathematics Kansas State University

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

nonabelian finite simple groups alternating groups (n ≥ 5) groups of Lie type 26 sporadic groups associated geometries Tits buildings sporadic geometries

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

nonabelian finite simple groups alternating groups (n ≥ 5) groups of Lie type 26 sporadic groups associated geometries Tits buildings sporadic geometries ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆

complexes of p-subgroups

group theory p-local structure algebraic topology mod-p cohomology classifying spaces ✁ ✁ ✁ representation theory Lefschetz modules ❆ ❆ ❆

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Outline of the Talk

1

Terminology and Notation

2

Background, History and Context

3

An Example: GL3(2)

4

Distinguished Collections of p-Subgroups

5

Lefschetz Modules for Distinguished Complexes

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Terminology and Notation: Groups

G is a finite group and p a prime dividing its order H.K denotes an extension of H by K pn denotes an elementary abelian group of order pn Op(G) is the largest normal p-subgroup in G Q a nontrivial p-subgroup of G H ≤ G is p-local subgroup if H = NG(Q) Q is p-radical if Q = Op(NG(Q)) Q is p-centric if Z(Q) ∈ Sylp(CG(Q))

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Terminology and Notation: Collections

Collection C family of subgroups of G closed under G-conjugation partially ordered by inclusion Subgroup complex |C| = ∆(C) simplices: σ = (Q0 < Q1 < . . . < Qn), Qi ∈ C isotropy group of σ: Gσ = ∩n

i=0NG(Qi)

fixed point set of Q: ∆(C)Q Let k be a field of characteristic p. The reduced Lefschetz kG-module:

• LG(∆(C); k) :=

dim(∆)

• i=−1

(−1)iCi(∆(C); k)

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Standard Collections of p-Subgroups

Brown Sp(G) nontrivial p-subgroups Quillen Ap(G) nontrivial elementary abelian p-subgroups Bouc Bp(G) nontrivial p-radical subgroups Quillen, 1978 Ap(G) ⊆ Sp(G) is homotopy equivalence

• LG(|Sp(G)|; k) is virtual projective module

Th´ evenaz, Webb, 1991 Ap(G) ⊆ Sp(G) ⊇ Bp(G) are equivariant homotopy equivalences

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Webb’s Alternating Sum Formula

Webb, 1987 assumes: ∆ is a G - simplicial complex ∆Q is contractible, Q any subgroup of order p proves:

• LG(∆; Zp) is virtual projective module
• Hn(G; M)p =

σ∈∆/G(−1)dim(σ)

Hn(Gσ; M)p

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Sporadic Geometries

first 2-local geometries constructed

Ronan and Smith, 1980 Ronan and Stroth, 1984

geometries with projective reduced Lefschetz modules

Ryba, Smith and Yoshiara, 1990

relate projectivity of the reduced Lefschetz module to p-local structure of the group

Smith and Yoshiara, 1997

connections with standard complexes and mod-2 cohomology for the 26 sporadic simple groups

Benson and Smith, 2004

Lefschetz characters for several 2-local geometries

Grizzard, 2007

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

An Example: GL3(2)

The Tits building: the extrinsic approach

e1 + e2 + e3 e2 e3 e1 e1 + e3 e1 + e2 e2 + e3

Fano Plane V = F3

2 = e1, e2, e3

p = e1 L = e1, e2 pL = (e1 ⊆ e1, e2)

Stabilizers

Gp =   1 ∗ ∗ ∗ ∗ ∗ ∗   GL =   ∗ ∗ ∗ ∗ ∗ ∗ 1   GpL =   1 ∗ ∗ 1 ∗ 1  

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The Tits building for GL3(2): the intrinsic approach

The quotient of the action of G on its building:

p

• L

The quotient of the action of G on its Bouc complex:

22

a

• D8
• 22

b

Barycentric subdivision of Tits building = Bouc complex Gp = S4 = 22

a.S3 = NG(22 a)

GL = S4 = 22

b.S3 = NG(22 b)

GpL = D8 = 21+2 = NG(D8) NG(22

a < D8) = NG(22 b < D8) = D8

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The reduced Lefschetz module of the Bouc complex = Steinberg module for GL3(2)

• LGL3(2)(|B2|) = −H1(∆) = −StGL3(2)

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The reduced Lefschetz module of the Bouc complex = Steinberg module for GL3(2)

• LGL3(2)(|B2|) = −H1(∆) = −StGL3(2)

Webb’s alternating formula for mod-2 cohomology:

0 → H∗(GL3(2); F2) → H∗(S4; F2) ⊕ H∗(S4; F2) → H∗(D8; F2) → 0

H∗(GL3(2); F2) = H∗(S4; F2) + H∗(S4; F2) − H∗(D8; F2)

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A 2-Local Geometry for Co3

G - Conway’s third sporadic simple group Co3 ∆ - subgroup complex with vertex stabilizers given below:

• P
• L
• M

Gp = 2.S6(2) GL = 22+63.(S3 × S3) GM = 24.L4(2)

Theorem (Maginnis and Onofrei, 2004)

The 2-local geometry ∆ for Co3 is homotopy equivalent to the complex of distinguished 2-radical subgroups | B2(Co3)|; 2-radical subgroups containing 2-central involutions in their centers.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Distinguished Collections of p-Subgroups

An element of order p in G is p-central if it lies in the center of a Sylow p-subgroup of G. Let Cp(G) be a collection of p-subgroups of G.

Definition

The distinguished collection Cp(G) is the collection of subgroups in Cp(G) which contain p-central elements in their centers.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Poset Homotopy

Two G-posets are G-homotopy equivalent if they are homotopy equivalent and the homotopies are G-equivariant. A poset C is conically contractible if there is a poset map f : C → C and an element x0 ∈ C such that x ≤ f(x) ≥ x0 for all x ∈ C. THEOREM [Th´ evenaz and Webb,1991]: Let C ⊆ D. Assume that for all y ∈ D the subposet C≤y = {x ∈ C | x ≤ y} is Gy-contractible. Then the inclusion is a G-homotopy equivalence.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Proposition (Maginnis and Onofrei, 2005)

The inclusion Ap(G) ֒ → Sp(G) is a G-homotopy equivalence.

Proof.

Let P ∈ Sp(G) and let Q ∈ Ap(G)≤P.

• P is the subgroup generated by the p-central elements in Z(P).

The subposet Ap(G)≤P is contractible via the double inequality: Q ≤ P · Q ≥ P The poset map Q → P · Q is NG(P)-equivariant.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Groups of Parabolic Characteristic p

G has characteristic p if CG(Op(G)) ≤ Op(G). G has local characteristic p if all p-local subgroups of G have characteristic p. G has parabolic characteristic p if all p-local subgroups which contain a Sylow p-subgroup of G have characteristic p.

Theorem (Maginnis and Onofrei, 2007)

Let G be a finite group of parabolic characteristic p. Then the collections Bp(G), Ap(G) and Sp(G) are G-homotopy equivalent.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Fixed Point Sets

Proposition (Maginnis and Onofrei, 2007 )

Let G be a finite group of parabolic characteristic p. Let z be a p-central element in G and let Z = z. Then the fixed point set | Bp(G)|Z is NG(Z)-contractible.

Proposition (Maginnis and Onofrei, 2007 )

Let G be a finite group of parabolic characteristic p. Let t be a noncentral element of order p and let T = t. Assume that Op(CG(t)) contains a p-central element. Then the fixed point set | Bp(G)|T is NG(T)-contractible.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Theorem (Maginnis and Onofrei, 2007 )

Assume G is a finite group of parabolic characteristic p. Let T = t with t an element of order p of noncentral type in G. Let C = CG(t). Suppose that the following hypotheses hold: Op(C) does not contain any p-central elements; The quotient group C = C/Op(C) has parabolic characteristic p. Then there is an NG(T)-equivariant homotopy equivalence | Bp(G)|T ≃ | Bp(C)|

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Sketch of Proof: The proof requires a combination of equivariant homotopy equivalences: | Bp(G)|T ≃ | Sp(G)|T ≃ | Sp(G)≤C

>T | ≃ |

Sp(G)≤C

>T |

≃ | Sp(G)≤C

>OC| ≃ |

Sp(G)≤C

>OC| ≃ |S| ≃ |

Sp(C)| ≃ | Bp(C)| Some of the notations used:

• Sp(G) = {p-subgroups of G which contain p-central elements},

C≤H

>P = {Q ∈ C | P < Q ≤ H},

OC = Op(C) and C = CG(t), S = {P ∈ Sp(G)≤C

>OC

• Z(P) ∩ Z(S) = 1,

for ST and S such that P ≤ ST ≤ S}, ST ∈ Sylp(C) which extends to S ∈ Sylp(G).

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Terminology from Representation Theory

k, a field of characteristic p, splitting field for G and all its subgroups; kG, the group algebra of G over the field of coefficients k; IndG

H (N) = kG ⊗kH N, the induced module,

for kH-module N and H ≤ G; IndG

Gx(k) ≃ k[X], permutation module,

for X a G-transitive set and x ∈ X. A kG-module M is relatively H-projective if M is a direct summand of an module induced from H. Let M be an indecomposable kG-module; V is a vertex of M if M is relatively V-projective, but is not relatively U-projective, for any proper subgroup U of V.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A block B of kG is an indecomposable two-sided ideal of kG. The defect group of a block B, is a subgroup D ≤ G with the property that δD = {(g, g); g ∈ D ≤ G} is a vertex of the k(G × G)-module B. The Green ring is a free abelian group generated by the isomorphism classes [M] of finitely generated indecomposable kG-modules. The ring structure is given by direct sums and k-tensor products. The reduced Lefschetz module, an element of the Green ring:

• LG(∆; k) =
• σ∈∆/G

(−1)|σ|IndG

Gσ(k) − k

THEOREM [Robinson, 1988 ]: The number of indecomposable summands of LG(∆; k) with vertex Q is equal to the number of indecomposable summands

• f

LNG(Q)(∆Q; k) with the same vertex Q.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

The Reduced Lefschetz Module for the Distinguished p-Radical Complex

Webb’s alternating sum formula holds for the distinguished Bouc complex:

• Hn(G; k) =
• σ∈|

Bp(G)|/G

(−1)dim(σ) Hn(Gσ; k) Assume that G has parabolic characteristic p. The vertices of the reduced Lefschetz module associated to Bp(G) are p-subgroups of pure noncentral type.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

A 2-Local Geometry for Fi22

G = Fi22 has parabolic characteristic 2. G has three conjugacy classes of involutions: CFi22(2A) = 2.U6(2), CFi22(2B) = (2 × 21+8

+

: U4(2)) : 2, CFi22(2C) = 25+8 : (S3 × 32 : 4). The class 2B is 2-central. ∆ is the 2-local geometry with vertex stabilizers: H1 = (2 × 21+8

+

: U4(2)) : 2 H2 = 25+8 : (S3 × A6) H3 = 26 : Sp6(2) H4 = 210 : M22 ∆ is G-homotopy equivalent to B2(Fi22).

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Theorem (Maginnis and Onofrei, 2007)

Let ∆ be the 2-local geometry for the Fischer group Fi22.

a The fixed point sets ∆2B and ∆2C are contractible. b The fixed point set ∆2A is equivariantly homotopy equivalent

to the building for the Lie group U6(2).

c There is precisely one nonprojective summand of the

reduced Lefschetz module, it has vertex 2A and lies in a block with the same group as defect group.

d As an element of the Green ring:

• LFi22(∆) = −PFi22(ϕ12) − PFi22(ϕ13) − 6ϕ15 − 12PFi22(ϕ16) − ϕ16.

Outline Terminology History GL3(2) Distinguished Collections Lefschetz Modules

Summary of the Talk

New collections of subgroups were introduced: emphasize the role of p-central elements; have homotopy properties similar to the standard collections of p-subgroups; are suited for cohomology computations; are related to the p-local geometries for the sporadic simple groups. Further objectives: determine the vertices of the reduced Lefschetz modules for other classes of groups;

• btain a general description of the indecomposable

summands of the reduced Lefschetz modules and their distribution into the blocks of the group ring.