Geometry, Topology and Dynamics Seminar
Kari Ragnarsson
DePaul University
Encoding fusion in the double Burnside ring
Abstract: Fusion systems were introduced by Puig as generalized models for
the p-local structure of a finite group. More concretely, a fusion system
on a finite p-group S is a category whose objects are the subgroups of S,
and whose morphism sets "look like" they are induced by conjugation in a
group that has S as Sylow subgroup. Fusion systems are studied in algebra,
where they arise naturally in block theory, and there is also an active
research program in algebraic topology on developing and studying
classifying spaces for fusion systems, resulting in the theory of p-local
finite groups.
In this talk I will present recent work, partly joint with Radu Stancu,
that gives a completely new way to look at fusion systems, via the double
Burnside ring. More precisely every fusion system has a ``characteristic
idempotent'' in the double Burnside ring, from which one can reconstruct
the fusion system. Furthermore, characteristic idempotents are exactly
those that satisfy a certain Frobenius reciprocity relation. Thus one
obtains a very surprising bijection between fusion systems and idempotents
satisfying Frobenius reciprocity. In addition to the immediate
consequences for p-local finite groups, this result settles long-standing
questions on the stable splitting of classifying spaces, and gives a
generalized version of the Adams--Wilkerson theorem.
Joint with the Algebra seminar
Monday October 20, 2008 at 4:00 PM in SEO 712