LMS/EPSRC Short Course: Fusion Systems
	July 30 to August 3, 2007 School of Mathematics The University of Birmingham Michael Aschbacher (Caltech) Radha Kessar (Aberdeen) Bob Oliver (Paris) also  Carles Broto (Barcelona) Markus Linckelmann (Aberdeen) Organisers: Chris Parker and Sergey Shpectorov

Introduction

The study of fusion systems brings together group theory, representation theory, and algebraic topology. Every finite group G and every p-subgroup S of G gives rise to a fusion system. When S is a Sylow p-subgroup of G, the corresponding fusion system F_S(G) satisfies various further properties. These properties were axiomatized by Puig, and when these additional conditions are satisfied the resulting fusion systems are called saturated fusion systems. The saturated fusion system F_S(G) controls the homological properties of the classifying space of G for the prime p. The interest in the classifying spaces is what provided the original motivation for Broto, Levi, and Oliver to define linking systems associated to fusion systems. Indeed the linking systems provide ``classifying spaces'' for abstract fusion systems: spaces which have many of the same properties as p-completions of classifying spaces of finite groups.

In representation theory, saturated fusion systems arise for each block and the role of the p-group is played by the associated defect group. The applications of fusion systems to the representation theory were the motivation for Puig's work. For many years, the only saturated fusion systems known were the ones arising from the defect groups of blocks (and of course from groups). However, more recently, examples of saturated fusion systems were constructed, which do not come from finite groups. Such fusion systems are called exotic. The most exciting of them is the so-called Solomon fusion system. This is a fusion system based on a 2-group of order 2¹⁰. A similar configuration was studied by Solomon when he showed that there were no simple groups which have centralizer of an involution isomorphic to Spin(7,q) with q odd. He discovered that the entire 2-local structure of the group in this case is compatible with the existence of a finite simple group, he could only reach a contradiction by studying the p-local structure of his potential groups for odd primes p. Years later Benson discovered that this compatible 2-local structure exists as an abstract fusion system which is not embeddable in a finite group (i.e. it is exotic). He additionally showed that the injective limit of such systems for increasing q leads to a topological space which is equivalent to the famous Dwyer-Wilkerson loop space BDI(4). As a mathematical object, fusion systems are based on amalgams of several p-local subgroups. In this respect, fusion systems are natural objects for finite group theorists to study. In particular, Aschbacher and Chermak gave an independent construction of the Solomon fusion system, exploiting the construction from finite group theory of so-called signalizer functors. They also showed that the Solomon fusion system contains subfusion systems which are related to several of the sporadic simple groups; namely, J₂, J₃ and Co₃.

Programme

The lecture series will start on Monday afternoon and finish on Friday. On Friday afternoon there will be two special lectures.

Here is the time-table

Course Overview

Course 1: Prof. Michael Aschbacher.  (Group Theory)

The group theory part of the course will focus on extending results in the local theory of finite groups to the more general setting of saturated fusion systems and p-local finite groups. We will discuss the exotic 2-local finite groups of Levi-Oliver/Benson-Solomon, constructed as fixed points of Frobenius maps on a 2-local compact group, realized as the fusion system of the free amalgamated product of algebraic groups, with linking system constructed via signalizer functors.

However, we will begin with basic results on saturated fusion systems due to Puig and BLO (Broto-Levi-Oliver). Then we will discuss the BCGLO (BLO+Castellana + Grodal) theorem that constrained fusion systems have models as the fusion system of a constrained finite group. Then we move to deeper material, such as the notion of a normal subsystem of a saturated fusion system, the generalized Fitting subsystem of such a system, etc. We will close with a discussion of the exotic 2-local examples.

Course 2: Dr Radha Kessar (Representation Theory)

Every block of a finite group algebra over a field of positive characteristic has an associated saturated fusion system the nature of which has a profound effect on the underlying representation theory. The course will begin with a general introduction--group algebras, Maschke's theorem, failure of complete reducibility in positive characteristic, p-modular systems, blocks. This will be followed by the core concepts-- Brauer homomorphism, defect groups, Brauer pairs, fusion systems over maximal Brauer pairs, proof of saturation. The relationships between the fusion system and other invariants of blocks, both known and conjectural will be studied relative projectivity, structure of blocks with central defect groups and extensions (Kulshammer-Puig structure theorem), Alperin's weight conjecture, alternating chain formulations, fusion category algebras, glueing problems. The course will also discuss the nature of fusion systems arising from blocks--principal blocks (Brauer's third main theorem). Are all block fusion systems non-exotic? Can one develop a Clifford theory for block fusion systems?

Course 3: Prof. Bob Oliver (Topology)

This series of talks will begin with background material including the nerve of a category. We will also discuss the p-completion construction due to Bousfield and Kan. We will define the classifying space for a saturated fusion system  F in terms of the p-completion of the nerve of a linking system associated to F.

When G is a finite group, the p-completion BG_p^ of the ordinary classifying space BG has certain unusual homotopy theoretic properties; properties which follow as consequences of the Sullivan conjecture in the form proved by Miller, Carlsson, and Lannes. For example, for any finite p-group P, the set of maps from BP to BG up to homotopy (up to continuous deformation) is in bijective correspondence with the set of homomorphisms from P to G up to conjugacy. The background to these results, and the analogous results for classifying spaces of fusion systems, will be discussed during these talks.

Finally, some examples of  "exotic" fusion systems and their classifying spaces will be described. The simplest examples are those due to Ruiz and Viruel, over the extraspecial group of order 7³ and exponent 7. Other examples which will be looked at include the 2-local fusion systems predicted by work of Solomon and constructed explicitly by Levi and Oliver, and their close connection with the Dwyer-Wilkerson space as predicted by Dave Benson. This also helps to motivate some generalizations of p-local finite groups with connections to compact Lie groups and p-compact groups.

Special Lectures

Professor Carles Broto: "Homotopy theory and p-local groups"

Professor Markus Linckelmann: "On control of fusion"

Accommodation and Entertainment 

Accommodation has been arranged in single ensuite rooms in the university's halls of residence.

Here's a website giving limited  information. Note that we have breakfast in a room called Fusion. That should be easy to remember. Here's the website. And here is another one.

Rebecca Waldecker has kindly put together an entertainment website for us. It's here.

Here are some maps and various directions.

More details will follow.

Registration 

The number of participants will be limited and those interested are encouraged to make an early application. An online application form is available from the London Mathematical Society.

The registration fee is 100 pounds which, for UK-based research students, includes the cost of course accommodation and meals. Participants must pay their own travel costs. EPSRC-supported students can expect that their registration fees and travel costs will be met by their departments from the EPSRC Research Training and Support Grant that is paid to universities with each studentship award.  Postgraduate students not studying at UK universities and Research Fellows are must to pay a registration fee of 250 pounds  in addition to the costs of their accommodation. The total cost will be 510 pounds. Such students are encouraged to contact the organizers directly.

Course Material 

Here will be a list of prerequisite reading for people attending the course.

Here are course notes for Professor Aschbacher's contribution to the summer school. 

Here are notes by Dr Kessar. (this is a dvi file) They give an introduction to block theory. 

Here are notes by Professor Linckelmann. These notes contain many of the standard results on fusion systems. 

Here is a survey that will offer valuable background material.

Important references include

Aschbacher, M. Finite group theory. Second edition. Cambridge Studies in
Advanced Mathematics, 10. Cambridge University Press, Cambridge,
2000. 

Broto, Carles; Levi, Ran; Oliver, Bob The theory of $p$-local groups: a survey. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, 51--84,
Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.

Broto, Carles; Levi, Ran; Oliver, Bob The homotopy theory of fusion
systems. J. Amer. Math. Soc. 16 (2003), no. 4, 779--856. This can be downloaded here. This appears as reference BL0 in Aschbacher's notes.

Further Information 

Further information is available from: Chris Parker and Sergey Shpectorov.