On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups - - PowerPoint PPT Presentation

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On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups - - PowerPoint PPT Presentation

Aug 13, 2023 169 likes •445 views

On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups Silvia Onofrei in collaboration with John Maginnis Kansas State University Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 1/15

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SLIDE 1

On Fixed Point Sets and Lefschetz Modules for Sporadic Simple Groups

Silvia Onofrei

in collaboration with John Maginnis

Kansas State University

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 1/15

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SLIDE 2

Terminology and Notation: Groups

G is a finite group and p a prime dividing its order Q a nontrivial p-subgroup of G Q is p-radical if Q = Op(NG(Q)) Q is p-centric if Z(Q) ∈ Sylp(CG(Q)) 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

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 2/15

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SLIDE 3

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 = ∆(C )Q Standard collections all subgroups are nontrivial Brown Sp(G) p-subgroups Quillen Ap(G) elementary abelian p-subgroups Bouc Bp(G) p-radical subgroups Bcen

p (G)

p-centric and p-radical subgroups Equivariant homotopy equivalences: Ap(G) ⊆ Sp(G) ⊇ Bp(G)

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 3/15

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SLIDE 4

Terminology and Notation: Lefschetz Modules

k field of characteristic p ∆ subgroup complex ∆/G the orbit complex of ∆ The reduced Lefschetz module alternating sum of chain groups

  • LG(∆;k) := ∑

|∆| i=−1(−1)iCi(∆;k)

element of Green ring of kG

  • LG(∆;k) = ∑σ∈∆/G(−1)|σ|IndG

Gσ k − k

  • for a Lie group in defining characteristic

LG(|Sp(G)|;k) is equal to the Steinberg module

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

  • Th´

evenaz (1987): LG(∆;k) is X -relatively projective X is a collection of p-subgroups ∆Q is contractible for every p-subgroup Q ∈ X

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 4/15

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SLIDE 5

Background, History and Context

if ∆Q is contractible for Q any subgroup of order p then LG(∆;Zp) is virtual projective module and Hn(G;M)p = ∑σ∈∆/G(−1)|σ| Hn(Gσ;M)p

Webb, 1987

sporadic geometries with projective reduced Lefschetz modules

Ryba, Smith and Yoshiara, 1990

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

Smith and Yoshiara, 1997

  • L(|Bcen

p |;k) is projective relative to the collection of p-subgroups

which are p-radical but not p-centric

Sawabe, 2005

connections between 2-local geometries and standard complexes for the 26 sporadic simple groups

Benson and Smith, 2008

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 5/15

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SLIDE 6

A 2-Local Geometry for Co3

G - Conway’s third sporadic simple group Co3 ∆ - standard 2-local geometry with vertex stabilizers given below:

  • P

Gp = 2.Sp6(2) GL = 22+63.(S3 ×S3) GM = 24.L4(2)

  • L
  • M

Theorem [MO] The 2-local geometry ∆ for Co3 is equivariant homotopy equivalent to the complex of distinguished 2-radical subgroups | B2(Co3)|;

2-radical subgroups containing 2-central involutions in their centers.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 6/15

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SLIDE 7

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.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 7/15

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SLIDE 8

A Homotopy Equivalence

Proposition [MO]

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

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15

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A Homotopy Equivalence

Proposition [MO]

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

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15

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A Homotopy Equivalence

Proposition [MO]

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

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.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 8/15

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SLIDE 11

The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ֒ → Sp(G) is a G-homotopy equivalence

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

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SLIDE 12

The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ֒ → Sp(G) is a G-homotopy equivalence Webb’s alternating sum formula holds for Bp(G)

H∗(G; LG(| Bp|;k)) = 0

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

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SLIDE 13

The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ֒ → Sp(G) is a G-homotopy equivalence Webb’s alternating sum formula holds for Bp(G)

H∗(G; LG(| Bp|;k)) = 0

Bcen

p

⊆ Bp ⊆ Bp

if G has parabolic characteristic p then Bp = Bcen

p

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

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SLIDE 14

The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ֒ → Sp(G) is a G-homotopy equivalence Webb’s alternating sum formula holds for Bp(G)

H∗(G; LG(| Bp|;k)) = 0

Bcen

p

⊆ Bp ⊆ Bp

if G has parabolic characteristic p then Bp = Bcen

p

| Bp(G)| is homotopy equivalent to the standard 2-local geometry for all but two (Fi23 and O′N) sporadic simple groups

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

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SLIDE 15

The Distinguished Bouc Collection

If G has parabolic characteristic p, then Bp(G) ֒ → Sp(G) is a G-homotopy equivalence Webb’s alternating sum formula holds for Bp(G)

H∗(G; LG(| Bp|;k)) = 0

Bcen

p

⊆ Bp ⊆ Bp

if G has parabolic characteristic p then Bp = Bcen

p

| Bp(G)| is homotopy equivalent to the standard 2-local geometry for all but two (Fi23 and O′N) sporadic simple groups

  • Bp(G) preserves the geometric interpretation of the points of the

geometry in cases where Bcen

p

does not

in Co3, the 2-central involutions (the points of the geometry) are 2-radical but not 2-centric

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 9/15

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SLIDE 16

Fixed Point Sets

Proposition 1 [MO] 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.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 10/15

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SLIDE 17

Fixed Point Sets

Proposition 1 [MO] 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 2 [MO] 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.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 10/15

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SLIDE 18

Fixed Point Sets

Theorem 3 [MO] 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)|

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 11/15

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SLIDE 19

Fixed Point Sets: Sketch of the Proof of Theorem 3

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).

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 12/15

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SLIDE 20

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, are 2-central CFi22(2C) = 25+8 : (S3 ×32 : 4)

∆ is the standard 2-local geometry for G, it is G-homotopy equivalent to B2(G) and has vertex stabilizers:

  • 6
  • 1

✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏

t

  • 5

10

H1 = (2×21+8

+

: U4(2)) : 2 H5 = 25+8 : (S3 ×A6) H6 = 26 : Sp6(2) H10 = 210 : M22

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 13/15

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A 2-Local Geometry for Fi22

Proposition 4 [MO] 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.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 14/15

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SLIDE 22

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of LG(∆) with vertex Q is equal to the number of indecomposable summands of

  • LNG(Q)(∆Q) with the same vertex Q.

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

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SLIDE 23

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of LG(∆) with vertex Q is equal to the number of indecomposable summands of

  • LNG(Q)(∆Q) with the same vertex Q.

the involutions 2B are central, Proposition 1 implies ∆2B is contractible O2(CG(2C)) contains 2-central elements, Proposition 2 implies that ∆2C is contractible ∆Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus LNG(Q)(∆Q) = 0

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

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SLIDE 24

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of LG(∆) with vertex Q is equal to the number of indecomposable summands of

  • LNG(Q)(∆Q) with the same vertex Q.

the involutions 2B are central, Proposition 1 implies ∆2B is contractible O2(CG(2C)) contains 2-central elements, Proposition 2 implies that ∆2C is contractible ∆Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus LNG(Q)(∆Q) = 0 no vertex of an indecomposable summand of LG(∆) contains an involution of type 2B or 2C

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

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SLIDE 25

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of LG(∆) with vertex Q is equal to the number of indecomposable summands of

  • LNG(Q)(∆Q) with the same vertex Q.

the involutions 2B are central, Proposition 1 implies ∆2B is contractible O2(CG(2C)) contains 2-central elements, Proposition 2 implies that ∆2C is contractible ∆Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus LNG(Q)(∆Q) = 0 no vertex of an indecomposable summand of LG(∆) contains an involution of type 2B or 2C CG(2A)/O2(CG(2A)) = U6(2), Theorem 3 implies that ∆2A is homotopy equivalent to the building for U6(2) ∆Q is contractible for any Q > 2A

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15

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SLIDE 26

Proof of the Proposition 4 Theorem [Robinson]: The number of indecomposable summands of LG(∆) with vertex Q is equal to the number of indecomposable summands of

  • LNG(Q)(∆Q) with the same vertex Q.

the involutions 2B are central, Proposition 1 implies ∆2B is contractible O2(CG(2C)) contains 2-central elements, Proposition 2 implies that ∆2C is contractible ∆Q is mod-2 acyclic for any 2-group Q containing an involution of type 2B or 2C (Smith theory) thus LNG(Q)(∆Q) = 0 no vertex of an indecomposable summand of LG(∆) contains an involution of type 2B or 2C CG(2A)/O2(CG(2A)) = U6(2), Theorem 3 implies that ∆2A is homotopy equivalent to the building for U6(2) ∆Q is contractible for any Q > 2A there is one nonprojective summand, it has vertex 2A

Silvia Onofrei (Kansas State University) Lefschetz modules for sporadic simple groups 15/15