FIXED POINT SETS AND LEFSCHETZ MODULES John Maginnis and Silvia - - PDF document

AMS Sectional Meeting Middle Tennessee State University, Murfreesboro, TN 3 - 4 November 2007

FIXED POINT SETS AND LEFSCHETZ MODULES John Maginnis and Silvia Onofrei Department of Mathematics, Kansas State University

• Abstract. The reduced Lefschetz modules associated to complexes of distinguished p-subgroups (those

subgroups which contain p-central elements in their centers) are investigated. A special class of groups, those of parabolic characteristic p, is analyzed in detail. We determine the nature of the fixed point sets of groups of order p. The p-central elements have contractible fixed point sets. Under certain hypotheses, the non-central p-elements have fixed points which are eqivariantly homotopy equivalent to the corresponding complex for a qutient of the centralizer. For the reduced Lefschetz module, the vertices of the indecomposable summands and the distribution of these summands into the p-blocks of the group ring are related to the fixed point sets.

• 1. Introduction, basic terminology, motivation and context

The present work investigates various properties of the reduced Lefschetz modules. The underlying simplicial complexes arise in a natural way from the group structure and are relevant to the cohomology and to the representation theory of the group. We are specially interested in those complexes which can be related to the p-local geometries for the sporadic groups. The best known example of a Lefschetz module is the Steinberg module of a Lie group G in defining

• characteristic. This module is irreducible and projective and it is the homology module of the associated

building. For finite groups in general, the Brown complex of inclusion chains of nontrivial p-subgroups has projective reduced Lefschetz module. For a Lie group in defining characteristic, this complex is G- homotopy equivalent to the building, and thus the Lefschetz module is equal to the corresponding Steinberg module. Webb [20, Theorem A′] assumed that the reduced Euler characteristics of the fixed point sets of subgroups of order p are zero and showed that the reduced Lefschetz module is projective in the Green ring of finitely generated ZpG-modules, with Zp the p-adic integers. Further, Webb [20, Theorem A] proved that the p-components of the cohomology of the group G with coefficients in a ZG-module M can be written as an alternating sum of the cohomology groups of the simplex stabilizers Gσ: Hn(G; M)p =

• σ∈∆/G

(−1)dim(σ)Hn(Gσ; M)p This result is known in the literature as Webb’s alternating sum formula. Ryba, Smith and Yoshiara [13] proved projectivity, via Webb’s method of contractible fixed point sets, for Lefschetz modules of 18 sporadic geometries. These are certain subgroup complexes admitting a flag-transitive action by a sporadic simple group, or such a geometry other than the building for Lie type groups. Some of the characters of these projective virtual modules and the decomposition into projective covers of simple modules were also determined. Smith and Yoshiara [17] approached the study of projective Lefschetz modules for sporadic geometries in a more systematic way. The results on projectivity are obtained via a homotopy equivalence with the Quillen complex of nontrivial elementary

1

2 FIXED POINT SETS AND LEFSCHETZ MODULES

abelian p-subgroups. It follows from their work that the sporadic groups of local characteristic p behave like the groups of Lie type in defining characteristic p; their reduced Lefschetz modules are projective. Further results on the Lefschetz modules associated to complexes of subgroups in sporadic simple groups are given in a recent book by Benson and Smith [1], who give homotopy equivalences for each sporadic group, between a particular 2-local geometry and the simplicial complex determined by a certain collection of 2-subgroups. They choose geometries for which Webb’s alternating sum formula is valid, but in many cases the reduced Lefschetz module is only acyclic, not projective. Our approach emphasizes the role of p-central elements. In [7], we defined certain collections of p- subgroups, which we call distinguished. These are subcollections of the standard collections of p- subgroups and which consist of those p-subgroups which contain p-central elements in their centers. In an attempt to better correlate the homotopical properties of p-subgroup complexes to the algebraic properties of their reduced Lefschetz modules, we initiated a systematical study of the fixed point sets

• f groups of order p acting on various complexes of distinguished p-subgroups; see [?]. At a first stage,

we studied groups G of parabolic characteristic p, a technical condition on the p-local structure of G which is satisfied by about half of the sporadic groups in characteristic 2 or 3. For ∆ the complex of distinguished p-radical subgroups, we show that the fixed point set of a p-central element is contractible. For an element t which is not of central type, the homotopy type of the corresponding fixed point set ∆t is determined by the group structure of CG(t). Under certain hypotheses, we proved that the fixed point set is equivariantly homotopy equivalent to the complex of distinguished p-radical subgroups for the quotient CG(t)/Op(CG(t)) of the centralizer. In [8] we combined various results on the homotopy properties of the distinguished complex of p-radical subgroups with techniques from modular representation theory in order to determine properties of reduced Lefschetz modules for sporadic groups. For odd primes, the distinguished p-radical complexes for several sporadic groups which have a Sylow p-subgroup of order p3, where p = 3, 5, 7, or 13, were analyzed in detail. Information on the structure of the corresponding Lefschetz modules is given, such as the distribution of indecomposable summands into blocks of the group ring. The vertices of the non- projective indecomposable summands are described. One interesting feature of a few of the examples is the existence of a non-projective summand of the reduced Lefschetz module lying in the principal block.

• 2. Notations, terminology and hypotheses

The Groups. Let G be a finite group and p a prime dividing its order. Let Q be a p-subgroup of G. Then Q is a p-radical subgroup if Q is the largest normal p-subgroup in its normalizer NG(Q). Next, Q is called p-centric if its center Z(Q) is a Sylow p-subgroup of CG(Q). A subgroup H of G is a p-local subgroup of G if there exists a nontrivial p-subgroup P ≤ G such that NG(P) = H. For a subgroup H

• f G, denote by Op(H) the largest normal p-subgroup of H.

The group G has characteristic p if CG(Op(G)) ≤ Op(G). If all p-local subgroups of G have character- istic p then G has local characteristic p. A parabolic subgroup of G is defined to be a subgroup which contains a Sylow p-subgroup of G. The group G has parabolic characteristic p if all p-local, parabolic subgroups of G have characteristic p. Examples of groups of local characteristic p are the groups of Lie type defined over fields of characteristic p, some of the sporadic groups (such as M22, M24, Co2 for p = 2). Any group of local characteristic p has parabolic characteristic p. Some examples of sporadic groups of parabolic characteristic p are: M12, J2, Co1 for p = 2, M12, J3, Co1 for p = 3, J2, Co1, Co2 for p = 5. Collections of p-subgroups. A collection C is a family of subgroups of G which is closed under conju- gation by G and it is partially ordered by inclusion; hence a collection is a G-poset.

FIXED POINT SETS AND LEFSCHETZ MODULES 3

The nerve |C| is the simplicial complex whose simplices are proper inclusion chains in C. This is the subgroup complex associated to C. If C is viewed as a small category, then |C| is the classifying space

• f C; the functor C → |C| assigns topological concepts to posets and categories.

For a simplex σ in |C| let Gσ denote its isotropy group. Also let CQ denote the elements in C fixed by Q, the corresponding subcomplex affords the action of NG(Q). To simplify the notation we will omit the | − |’s. In what follows Sp(G) will denote the Brown collection of nontrivial p-subgroups and Bp(G) the Bouc collection of nontrivial p-radical subgroups. The inclusion Bp(G) ⊆ Sp(G) is a G-homotopy equivalence [19, Thm.2]. Let Bcen

p

(G) be the collection of nontrivial p-radical and p-centric subgroups. This collection is not in general homotopy equivalent with Sp(G). Distinguished collections of p-subgroups. An element of order p in G is called p-central if it lies in the center of a Sylow p-subgroup of G. For Cp(G) a collection of p-subgroups of G denote by Cp(G) the collection of subgroups in Cp(G) which contain p-central elements in their centers. We call Cp(G) the distinguished Cp(G) collection. We shall refer to the subgroups in Cp(G) as distinguished subgroups. Also, denote Cp(G) the collection of subgroups in Cp(G) which contain p-central elements. Obviously Cp(G) ⊆ Cp(G) ⊆ Cp(G). The reduced Lefschetz module. Let k denote a field of characteristic p. The reduced Lefschetz module is the virtual module in the Green ring A(G) of finitely generated kG-modules:

• LG(∆, k) =
• σ∈∆/G

(−1)dim(σ)IndG

Gσk − k

where ∆/G denotes the orbit complex of ∆. The reduced Lefschetz module associated to a subgroup complex ∆ is not always projective; however, it was shown by Th´ evenaz [18, Theorem 2.1] that this virtual module turns out to be, in many cases, projective relative to a collection of very small order p-groups.

• LG(∆, k) is X-relatively projective,

where X is a collection of p-subgroups such that ∆Q is acyclic (for example contractible) for every p-subgroup Q which is not in X. Under the Webb condition for the complex ∆ the reduced Lefschetz module is virtual projective, so H∗(G; L(∆)) = 0 and this idea is behind Webb’s proof of the alternating sum formula, which is reinterpreted in modern language as “normalizer sharpness”. However, for collections which are “ample” but not homotopy equivalent to the Brown complex, the reduced Lefschetz module is not projective but it is acyclic, in the sense that H∗(G, L(∆)) = 0. All the collections mentioned in this talk have this property: they are acyclic although their Lefschetz modules are usually not projective.

• 3. Fixed point sets of p-elements

This section contains the three main results presented in this talk, they describe the structure of the fixed point sets, under the action of elements of order p. The group G is assumed to have parabolic characteristic p and the underlying subgroup complex is Bcen

p

(G) = Bp(G). Regarding Bcen

p

(G) as a distinguished collection allows us to use certain features, such as equivariant homotopy equivalences with other collections, which are very useful in proving the results. Proposition 3.1. Let G be a finite group of parabolic characteristic p. Set Z = z with z a p-central element in G. The fixed point set BZ

p (G) is NG(Z)-contractible.

Proposition 3.2. Let G be a finite group of parabolic characteristic p. Let t be a noncentral element

• f order p. Assume that Op(CG(T)) ∈

Sp(G). Then the fixed point set BT (G) is contractible.

4 FIXED POINT SETS AND LEFSCHETZ MODULES

Theorem 3.3. Assume G is a finite group of parabolic characteristic p. Set T = t with t an element

• f order p of noncentral type in G. Suppose that the following hypotheses hold:

(1) C = CG(T) does not have characteristic p; (2) The quotient group C = CG(T)/Op(CG(T)) has parabolic characteristic p. There is an NG(T)-equivariant homotopy equivalence BT

p (G) ≃

Bp(C).

• Proof. The proof of the theorem is quite technical and several pages long and recquires a combination
• f several equivariant homotopy equivalences:
• BT

p (G) ≃

ST

p (G) ≃

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:

• C≤H

>P = {Q ∈ C | P < Q ≤ H},

• OC = Op(C) the largest normal p-subgroup in C = CG(T);
• S = {P ∈

Sp(G)≤C

>OC

• Z(P) ∩ Z(S) = 1, for some ST and S with P ≤ ST ≤ S}

The proof relies on the following:

• results and techniques which use equivariant poset maps (due to Thevenaz and Webb);
• constructing strings of equivariant poset maps (most of which are homotopical equivalent to

the identity map);

• the p-local structure of the group G, in particular properties of groups of characteristic p and

local characteristic p;

• the closure property of the

Sp(G), which says that every p-overgroup of an element in Sp(G) is also an element of Sp(G).

• 4. The Lefschetz module associated to the complex of distinguished p-radical

subgroups The collection of nontrivial p-centric and p-radical subgroups Bcen

p

(G) is relevant to both modular representation theory and mod-p cohomology of G. This collection is normalizer sharp, in Dwyer’s sense, which means that there is an alternating sum formula for the cohomology of the group, written in terms of the normalizers of the p-chains in this complex. For many of the sporadic groups and p = 2, a standard sporadic 2-local geometry is equivariantly homotopy equivalent to the complex of p-radical and p-centric subgroups. The reduced Lefschetz module LG(Bcen

p

(G)) is acyclic, that is H∗(G, LG(Bcen

p

(G))) = 0. Sawabe [14, Section 4] showed that its reduced Lefschetz module LG(∆(Bcen

p

(G)), k) is projective rela- tive to the collection of p-subgroups which are p-radical but not p-centric. The vertices of its indecom- posable summands are restricted and an upper bound to their orders is determined [15, Prop. 5]; this upper bound equals the largest order of a p-subgroup which is p-radical but not p-centric. Sawabe also showed (under certain assumptions) that the maximal subgroups which are p-radical but not p-centric are indeed among the vertices of LG(∆(Bcen

p

(G)), k); [16]. If the group G has parabolic characteristic p then Bcen

p

(G) = Bp(G). Thus, our results on the homotopy type of the fixed point sets in groups of parabolic characteristic, lead to the conclusion that the vertices

• f the reduced Lefschetz module are subgroups of pure noncentral type.

FIXED POINT SETS AND LEFSCHETZ MODULES 5

• 5. Examples and Lefschetz modules

We will discuss three examples, and give an application to modular representation theory. Recall that if a group G acts on a simplicial complex ∆, we can construct the virtual Lefschetz module by taking the alternating sum of the vector spaces (over a field of characteristic p) spanned by the chains. To

• btain the reduced Lefschetz module, subtract the trivial one dimensional representation. Information

about fixed point sets leads to details about the vertices of indecomposable summands of this module. We will make repeated use of the following: Theorem A[Burry, Carlson, Puig, Robinson]: Let C be a collection of p-subgroups of G. Then the number of indecomposable summands of LG(|C|; k) with vertex Q is the same as the number of inde- composable summands of LNG(Q)(|CQ|; k) with vertex Q. The Fischer group G = Fi22 and p = 2 We begin with the sporadic simple Fischer group Fi22, which has parabolic characteristic 2 and has three conjugacy classes of involutions, denoted 2A, 2B and 2C in the Atlas. The class 2B is 2-central. Their centralizers are CFi22(2A) = 2.U6(2), CFi22(2B) = (2 × 21+8

+

: U4(2)) : 2 and CFi22(2C) = 25+8 : (S3 × 32 : 4). Proposition 5.1. Let ∆ be the standard 2-local geometry for the Fischer group G = 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. The Monster and p=13 Let G be the Monster M and let p = 13. Then the fixed point set ∆t consists of 144 contractible components, equivalent to the set of Sylow subgroups Syl13(L3(3)). LG(∆) contains three nonprojective summands, all having vertex 13 = t. These three summands lie in three different blocks; two lie in blocks with the same group 13 = t as defect group, but the third summand lies in the principal block. The Monster group M has two classes of elements of order 13, type 13A and 13B (the latter are 13-central). The normalizers are NM(13A) = (13 : 6 × L3(3)).2 and NM(13B) = NM(131+2) = 131+2 : (3 × 4.S4). There is one class of purely central elementary abelian 13B2. The groups 13A, 13B2, and 131+2 are 13-radical, and the last two are distinguished. Thus ∆ is a graph with two types

• f vertices; each Sylow contains six copies of 13B2, and each elementary abelian 13B2 lies in 14 Sylow
• subgroups. Note that 13A ∈ NM(131+2) = NM(13B) iff 13B ∈ CM(13A) = 13 × L3(3). Thus the fixed

point set ∆13A contains 144 vertices corresponding to Sylow subgroups. For each such Sylow, the 13A also normalizes its six elementary abelian 13B2, and if 13A ∈ NM(13B2), then they generate a Sylow 131+2 = 13A, 13B2. The group L3(3) has six blocks at the prime 13, of defects 1,0,0,0,0,0. The action of L3(3) on its Sylow 13-subgroups corresponds to the induced character IndL3(3)

13:3 (1a) = 1a+13a+16abcd+27a+39a. Note

that 13a is projective and lies in block two, and 39a is projective and lies in block six. The projective cover PM(16) = 16abcd+27a lies in the principal block. This implies that the reduced Lefschetz module

• LM(∆) contains three indecomposable summands with vertex 13 = 13A. Two of these summands lie

in two blocks, having the same group as defect group. The third summand lies in the principal block.

• 6. Concluding Remarks

Let ∆ denote the complex of distinguished p-radical subgroups in a group G and set L = LG(∆, k), the associated reduced Lefschetz module. Assuming that G has parabolic characteristic p. If in addition

6 FIXED POINT SETS AND LEFSCHETZ MODULES

the centralizer of every non-central p-element has a component H which is a Lie group in characteristic p or an extension of such a group, then the reduced Lefschetz module is acyclic. The non-projective indecomposable summands lie in non-principal blocks of kG and they are extensions of the Steinberg module of the component H. Further, if V is a defect group of kG and t ∈ Z(V ) then V is a defect group of CG(t). Under the parabolic characteristic assumption on G, the defect groups of the non- principal blocks of kG are defect groups of kCG(t) for p-elements of non-central type. Therefore there is a direct relation between the vertices of L and the defect groups of the group ring kG. References

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