Subgroup Complexes and their Lefschetz Modules

Transcription

1 Subgroup Complexes and their Lefschetz Modules Silvia Onofrei Department of Mathematics Kansas State University

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

3 nonabelian finite simple groups alternating groups (n 5) groups of Lie type 26 sporadic groups complexes of p-subgroups algebraic topology mod-p cohomology classifying spaces group theory p-local structure associated geometries Tits buildings sporadic geometries representation theory Lefschetz modules

4 Outline of the Talk 1 Terminology and Notation 2 Background, History and Context 3 An Example: GL 3 (2) 4 Distinguished Collections of p-subgroups 5 Lefschetz Modules for Distinguished Complexes

5 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 p n denotes an elementary abelian group of order p n O p (G) is the largest normal p-subgroup in G Q a nontrivial p-subgroup of G H G is p-local subgroup if H = N G (Q) Q is p-radical if Q = O p (N G (Q)) Q is p-centric if Z (Q) Syl p (C G (Q))

6 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 = (Q 0 < Q 1 <... < Q n ), Q i C isotropy group of σ: G σ = n i=0 N G(Q i ) fixed point set of Q: (C) Q Let k be a field of characteristic p. The reduced Lefschetz kg-module: L G ( (C); k) := dim( ) ( 1) i C i ( (C); k) i= 1

7 Standard Collections of p-subgroups Brown S p (G) nontrivial p-subgroups Quillen A p (G) nontrivial elementary abelian p-subgroups Bouc B p (G) nontrivial p-radical subgroups Quillen, 1978 A p (G) S p (G) is homotopy equivalence L G ( S p (G) ; k) is virtual projective module Thévenaz, Webb, 1991 A p (G) S p (G) B p (G) are equivariant homotopy equivalences

8 Webb s Alternating Sum Formula Webb, 1987 assumes: proves: is a G - simplicial complex Q is contractible, Q any subgroup of order p LG ( ; Z p ) is virtual projective module Ĥ n (G; M) p = σ /G ( 1)dim(σ)Ĥn (G σ ; M) p

9 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

10 An Example: GL 3 (2) The Tits building: the extrinsic approach e 1 e 1 + e 2 e 1 + e 3 e 1 + e 2 + e 3 e 2 e 2 + e 3 e 3 Fano Plane V = F 3 2 = e 1, e 2, e 3 p = e 1 L = e 1, e 2 pl = ( e 1 e 1, e 2 ) Stabilizers 1 1 G p = 0 G L = G pl =

11 The Tits building for GL 3 (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: 2 2 a D b Barycentric subdivision of Tits building = Bouc complex G p = S 4 = 2 2 a.s 3 = N G (2 2 a) G L = S 4 = 2 2 b.s 3 = N G (2 2 b ) G pl = D 8 = = N G (D 8 ) N G (2 2 a < D 8 ) = N G (2 2 b < D 8) = D 8

12 The reduced Lefschetz module of the Bouc complex = Steinberg module for GL 3 (2) L GL3 (2)( B 2 ) = H 1 ( ) = St GL3 (2)

13 The reduced Lefschetz module of the Bouc complex = Steinberg module for GL 3 (2) L GL3 (2)( B 2 ) = H 1 ( ) = St GL3 (2) Webb s alternating formula for mod-2 cohomology: 0 H (GL 3 (2); F 2 ) H (S 4 ; F 2 ) H (S 4 ; F 2 ) H (D 8 ; F 2 ) 0 H (GL 3 (2); F 2 ) = H (S 4 ; F 2 ) + H (S 4 ; F 2 ) H (D 8 ; F 2 )

14 A 2-Local Geometry for Co 3 G - Conway s third sporadic simple group Co 3 - subgroup complex with vertex stabilizers given below: P L M G p = 2.S 6 (2) G L = (S 3 S 3 ) G M = 2 4.L 4 (2) Theorem (Maginnis and Onofrei, 2004) The 2-local geometry for Co 3 is homotopy equivalent to the complex of distinguished 2-radical subgroups B 2 (Co 3 ) ; 2-radical subgroups containing 2-central involutions in their centers.

15 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 C p (G) be a collection of p-subgroups of G. Definition The distinguished collection Ĉp(G) is the collection of subgroups in C p (G) which contain p-central elements in their centers.

16 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 x 0 C such that x f (x) x 0 for all x C. THEOREM [Thévenaz and Webb,1991]: Let C D. Assume that for all y D the subposet C y = {x C x y} is G y -contractible. Then the inclusion is a G-homotopy equivalence.

17 Proposition (Maginnis and Onofrei, 2005) The inclusion Âp(G) Ŝp(G) is a G-homotopy equivalence. Proof. Let P Ŝp(G) and let Q Âp(G) P. P is the subgroup generated by the p-central elements in Z (P). The subposet  p (G) P is contractible via the double inequality: Q P Q P The poset map Q P Q is N G (P)-equivariant.

18 Groups of Parabolic Characteristic p G has characteristic p if C G (O p (G)) O p (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 B p (G), Âp(G) and Ŝp(G) are G-homotopy equivalent.

19 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 B p (G) Z is N G (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 O p (C G (t)) contains a p-central element. Then the fixed point set B p (G) T is N G (T )-contractible.

20 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 = C G (t). Suppose that the following hypotheses hold: O p (C) does not contain any p-central elements; The quotient group C = C/O p (C) has parabolic characteristic p. Then there is an N G (T )-equivariant homotopy equivalence B p (G) T B p (C)

21 Sketch of Proof: The proof requires a combination of equivariant homotopy equivalences: B p (G) T Ŝp(G) T Ŝp(G) C >T S p (G) C >T S p (G) C >O C Ŝp(G) C >O C S Ŝp(C) B p (C) Some of the notations used: S p (G) = {p-subgroups of G which contain p-central elements}, C H >P = {Q C P < Q H}, O C = O p (C) and C = C G (t), C S = {P Ŝp(G) >O C Z (P) Z (S) 1, for S T and S such that P S T S}, S T Syl p (C) which extends to S Syl p (G).

22 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; Ind G H (N) = kg kh N, the induced module, for kh-module N and H G; Ind G G x (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.

23 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: L G ( ; k) = ( 1) σ IndG G σ (k) k σ /G THEOREM [Robinson, 1988 ]: The number of indecomposable summands of L G ( ; k) with vertex Q is equal to the number of indecomposable summands of L NG (Q)( Q ; k) with the same vertex Q.

24 The Reduced Lefschetz Module for the Distinguished p-radical Complex Webb s alternating sum formula holds for the distinguished Bouc complex: Ĥ n (G; k) = ( 1) dim(σ) Ĥ n (G σ ; k) σ B p(g) /G Assume that G has parabolic characteristic p. The vertices of the reduced Lefschetz module associated to B p (G) are p-subgroups of pure noncentral type.

25 A 2-Local Geometry for Fi 22 G = Fi 22 has parabolic characteristic 2. G has three conjugacy classes of involutions: C Fi22 (2A) = 2.U 6 (2), C Fi22 (2B) = ( : U 4(2)) : 2, C Fi22 (2C) = : (S : 4). The class 2B is 2-central. is the 2-local geometry with vertex stabilizers: H 1 = ( : U 4(2)) : 2 H 2 = : (S 3 A 6 ) H 3 = 2 6 : Sp 6 (2) H 4 = 2 10 : M 22 is G-homotopy equivalent to B 2 (Fi 22 ).

26 Theorem (Maginnis and Onofrei, 2007) Let be the 2-local geometry for the Fischer group Fi 22. 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 U 6 (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: L Fi22 ( ) = P Fi22 (ϕ 12 ) P Fi22 (ϕ 13 ) 6ϕ 15 12P Fi22 (ϕ 16 ) ϕ 16.

27 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; obtain a general description of the indecomposable summands of the reduced Lefschetz modules and their distribution into the blocks of the group ring.