GRA Workshop 1

Last week was the introductory/instructional workshop for the Isaac Newton Institute’s six-month programme on Groups, Representations and Applications.

We were thrown in the deep end right at the start. The first two talks were on what were claimed to be two of the major themes of the programme, representations of reductive groups, and fusion systems. Neither of these are things that are within my comfort zone, and I felt considerably out of my depth for the first couple of days. Later things got better, with very nice talks by Colva Roney-Dougal on Aschbacher’s Theorem, Eamonn O’Brien on algorithms for matrix groups, and Martin Liebeck on subgroup structure of almost simple groups, to name just a few.

Anyway, after the first day, I felt a bit sorry for some of the beginning PhD students, and offered them a one-hour crash course on ordinary representation theory, a subject which in my view underlies much of what went on. (At St Andrews, for example, we have no module on representation theory on the books, though students can take it by independent study.)

It may be that these notes will be of wider interest, so I have uploaded them here. They should appear among my lecture notes.

A bit about fusion systems, which are one of the coming topics in group theory. This is my understanding, which might be quite wrong. In the Classification of Finite Simple Groups, it was often necessary to consider a p-local subgroup H of a finite simple group G (the normaliser of a non-trivial p-subgroup P of G), where p is prime (especially in the case p = 2). Now there may be additional conjugacies of subgroups of P not induced by elements of H. A fusion system is an axiomatic way of describing this, as a category whose objects are the subgroups of P and whose morphisms are conjugations induced by elements of G. Of course, once something like this is axiomatised, it is possible (and indeed happens, though it seems to be rare) that there exotic fusion systems not arising in the manner described. Now if we replace fusion arguments in subgroups of G by abstract fusion system arguments, some advantages appear. In particular, information about centralisers causes quite some difficulty in the proof of CFSG, and it is hoped that this can be avoided using fusion systems, since centralisers are invisible.

In addition, fusion systems appear in other parts of mathematics. Markus Linckelmann showed us how to associate a fusion system to a block of a modular group algebra, and Radha Kessar hinted that they also arise in a topological context.

Perhaps I have completely misunderstood, so please don’t take the above too seriously. If I have missed the point, perhaps the practitioners didn’t explain it clearly enough?