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University of California Santa Cruz Fusion Systems And UNIVERSITY OF CALIFORNIA SANTA CRUZ FUSION SYSTEMS AND BISET FUNCTORS VIA GHOST ALGEBRAS A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Shawn Michael O’Hare June 2013 The Dissertation of Shawn Michael O’Hare is approved: Professor Robert Boltje, Chair Professor Geoff Mason Professor Martin Weissman Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Shawn Michael O’Hare 2013 Table of Contents Abstract v Dedication vi Acknowledgments vii Introduction1 Some Notation . .4 1 Background7 1.1 Fusion System Basics . .7 1.2 Bisets . 10 1.3 Double Burnside Groups . 15 1.4 Two Ghost Groups . 17 1.5 Subgroups of Burnside Groups . 21 1.6 Biset Categories . 24 2 Biset Categories 27 2.1 A Special Class of Groups . 28 2.2 Fusion Preserving Isomorphisms . 31 3 Characteristic Idempotents 36 3.1 Calculating the Characteristic Idempotent . 36 3.2 Bideflation of Characteristic Idempotents . 42 4 A Generalized Burnside Functor 47 4.1 Pseudo-rings . 47 Condensation and Decondensation . 49 4.2 Decondensation of the Burnside Functor . 51 4.3 The Action of Elementary Subgroups . 53 4.4 Subfunctors . 57 References 67 iii A Single Burnside Rings 70 A.1 Sets with a Group Action . 70 A.2 Operations on Sets with a Group Action . 76 A.3 The Single Burnside Ring . 78 Notation 86 Index 88 iv Abstract Fusion Systems and Biset Functors via Ghost Algebras by Shawn Michael O’Hare In this work we utilize ghost groups of Burnside groups introduced by Boltje and Danz in order to investigate fusion systems of finite groups, double Burnside modules, and biset functors. We give an expression for the coefficients of the characteristic idempotent !F associated to an arbitrary fusion system F, and demonstrate that bideflation does not generally preserve this idempotent when F is unsaturated. Motivated by the theory of p-completed classifying spaces, we study when two groups are isomorphic in the left-free p-local biset category B¡p , and prove that two groups G and H are isomorphic in this category when there exists an isomorphism between their Frobenius p-fusion systems. Finally, we consider a process mirroring Green’s theory of idempotent condensation and demonstrate that a generalized Burnside functor is the decondensation of the usual Burnside functor, and that this decondensation preserves the subfunctor lattice. v To my family, for their encouragement and support. vi Acknowledgments First and foremost I am indebted to Robert Boltje, without whose insight and patience this thesis would not exist. I have learned a tremendous amount from each of my committee members, and they have helped me develop as a mathematician. Martin Flashman’s enthusiasm for the subject convinced me to pursue it in earnest. David Kinberg has my profound thanks for recognizing and fostering my intellectual curiosity when I was a teenager. I am also grateful for my friends in Santa Cruz, many of them also colleagues in the Mathematics Department. They have enriched my life to an ineffable degree. My early collaborations with Corey Shanbrom instilled me with the requisite tenacity to come as far as I have. Rob Laber and David Deconde have helped me expand my knowledge base and focus the direction of my future studies. I am in debt to Victor Dods and Wyatt Howard for their many hours of stimulating conversation. vii Introduction Double Burnside groups serve as the common core linking together all the objects we study in this thesis. For a finite group G, the set of isomorphism classes of finite G-sets is a commutative monoid under the coproduct of sets, and the Grothendieck group B(G) of this monoid is enriched with a commutative ring structure via the direct product of sets. We call B(G) the single Burnside ring of G. Given another finite group H and commutative ring R, we define the double Burnside R-module RB(G; H) to be R ⊗Z B(G × H). There is a natural correspondence between (G × H)-sets and (G; H)-bisets, which are sets endowed with a left G-action and a right H-action that commute, given by viewing a left (G × H)-set X as a (G; H)-biset via gxh := (g; h−1)x for g 2 G, h 2 H, x 2 X. When G = H, the tensor product of bisets induces a generally non-commutative ring structure on B(G; G), which we then call the double Burnside ring of G. While the single and double Burnside rings of G encode much representation theoretic information about G, their multiplicative structures can be complicated. One successful remedy is to embed the Burnside rings into other rings, called ghost rings, with more tractable multiplicative structures. This is classically done by taking marks, i.e., computing fixed points. Our study of fusion systems and biset functors primarily exploits ghost groups for the double Burnside group introduced by Boltje and Danz in [BD12; BD13]. Fusion systems are of interest to abstract group theorists, modular representa- tion theorists, and topologists. There is hope that the techniques involved in classifying saturated fusion systems could streamline the classification of finite simple groups. As- sociated to every p-block of a finite group is a saturated fusion system, and these fusion systems provide a nice context in which to investigate aspects of modular representation theory. The Martino-Priddy conjecture [MP96], proved by Oliver [BLO03; Oli04; Oli06], asserts that two groups have homotopy equivalent p-completed classifying spaces if and 1 only if their Frobenius p-fusion systems are equivalent. Ragnarsson and Stancu [RS13] create a bijection between saturated fusion systems on a p-group S and the set of characteristic idempotents in the bifree dou- ∆ ble Burnside Z(p)-algebra Z(p)B (S; S). Boltje and Danz [BD12] construct a ghost ring B~∆(S; S) for the bifree double Burnside ring B∆(S; S), a mark homomorphism ρ: B∆(S; S) ! B~∆(S; S), and are able to extend the correspondence in [RS13] to the set of all fusion systems on S and a set of certain explicitly described idempotents in ~∆ ∆ the rational bifree ghost algebra QB (S; S). While the image of Z(p)B (S; S) under ρ ~∆ ~∆ is contained in Z(p)B (S; S), the idempotents in Z(p)B (S; S) do not necessarily corre- spond to saturated fusion systems. One of our initial hopes was to furnish a completely algebraic proof that a fusion system on S is saturated if and only if its corresponding ∆ characteristic idempotent !F is an element of Z(p)B (S; S), which was initially proved by topological means in [RS13]. As of this writing, our goal remains unachieved. The basic theory of saturated fusion systems can be reformulated in terms of characteristic idempotents [RS13, Section 8]. In particular, the characteristic idempo- tents of saturated fusion systems are well-behaved with respect to factoring, in the sense that that if F is a saturated fusion system on S and F=T is a quotient system over a strongly F-closed subgroup T , then the characteristic idempotents of F and F=T are S related by bideflation, i.e., BidefT (!F ) = !F=T . We show in Section 3.2 that this result does not extend generally by constructing an unsaturated fusion system F and quotient S system F=T such that BidefT (!F ) 6= !F=T . A convenient setting for our studies are various subcategories of the R-biset category BR, and categories related to them by means of ghost groups. The category BR has as objects the class of finite groups (or, to get an equivalent category, a transversal for the isomorphism classes of finite groups) and for two objects G and H we define HomBR (G; H) := RB(H; G). Morphism composition is induced from the tensor product of bisets. Both [Rag07, Theorem A] and [RS13, Section 9] relate the stable homotopy of classifying spaces of finite groups to the left-free p-local biset category B¡p , which is a subcategory of the Z(p)-biset category with the same object class, but whose morphisms are generated by transitive bisets corresponding to p-groups. To better understand stable equivalences, it would be useful to determine whether B¡p has the same isomorphism classes as the bifree p-local biset category B∆p , a subcategory of B¡p whose morphisms are generated by bifree bisets. In Chapter2 we use the N-grading of ghost algebras 2 introduced in [BD12, Section 6] to explore when two groups are isomorphic in B¡p . In Section 2.2 we prove that if there is a fusion preserving isomorphism between the Frobenius p-fusion systems of G and H, then G and H are isomorphic in B∆p , further relating fusion systems to stable homotopy of classifying spaces. Many natural operations that occur in the representation theory of finite groups– for example, restriction and induction–arise as functors involving bisets. As these opera- tions appear in a variety of contexts, such as group cohomology, the algebraic K-theory of group rings, and algebraic number theory, the abstract study of these operations is sufficiently motivated. The formalism of operations like restriction and induction is en- coded by biset functors, which are R-bilinear functors from some subcategory of BR to the category of R-modules. Their theory was the main tool in the classification of endo-permutation modules for a p-group [Bou06]. Biset functors have also been lever- aged successfully to determine the unit group of the single Burnside ring of a p-group [Bou07].
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