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)


Carles Broto (Barcelona)
Markus Linckelmann (Aberdeen)

Andrew Chermak  (Kansas State)

Organisers: Chris Parker and Sergey Shpectorov


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 BGp^ 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 Andrew Chermak: "Morphisms of p-local finite 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.

There will be a coach excursion to Stratford upon Avon on Wednesday Afternoon.

Here are some maps and various directions.


More details will follow.

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 by Boto, Levi and Oliver that will offer valuable background material.

Here are some notes about fusion in finite groups by Paul Flavell. These might give those of you with a limited background in group theory an idea where many of the results in fusion systems first arose.

Important references include

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

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.


Page last modified: June, 2007