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Fixed Point Sets and Lefschetz Modules

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Fixed Point Sets and Lefschetz Modules 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 A0] 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: n X dim(σ) n H (G; M)p = (−1) H (Gσ; M)p σ∈∆/G 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 of 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 hti ∆ 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 of 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 of C; the functor C → |C| assigns topological concepts to posets and categories. Q For a simplex σ in |C| let Gσ denote its isotropy group. Also let C 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 cen [19, Thm.2]. Let Bp (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 Cbp(G) the collection of subgroups in Cp(G) which contain p-central elements in their centers. We call Cbp(G) the distinguished Cp(G) collection. We shall refer to the subgroups in Cbp(G) as distinguished subgroups. Also, denote Cep(G) the collection of subgroups in Cp(G) which contain p-central elements. Obviously Cbp(G) ⊆ Cep(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: X dim(σ) G LeG(∆, k) = (−1) IndGσ k − k σ∈∆/G 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.
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