Burnside rings of fusion systems

In this thesis we study the interactions between saturated fusion systems and group actions of the underlying p-groups. For a saturated fusion system F on a finite p-group S we construct the Burnside ring of F in terms of the finite S-sets whose actions respect the structure of the fusion system, and we produce a basis for the Burnside ring that shares properties with the transitive sets for a finite group.

We construct a transfer map from the p-local Burnside ring of the underlying p-group S to the p-local Burnside ring of F. Using such transfer maps, we give a new explicit construction of the characteristic idempotent of F -- the unique idempotent in the p-local double Burnside ring of S satisfying properties of Linckelmann and Webb. We describe this idempotent both in terms of fixed points and as a linear combination of transitive bisets.

Additionally, using fixed points we determine the map of Burnside rings given by multiplication with the characteristic idempotent, and we show that this map is the transfer map previously constructed. Applying these results, we show that for every saturated fusion system the ring generated by all (non-idempotent) characteristic elements in the p-local double Burnside ring is isomorphic to the p-local Burnside ring of the fusion system, and we disprove a conjecture by Park-Ragnarsson-Stancu on the composition product of fusion systems.

Principal supervisor: Prof. Jesper Grodal

Assessment committee:
Chairman, Prof. Jesper Michael Møller, Institut for Matematiske Fag

Prof. Serge Bouc, Université de Picardie

Prof. Bob Oliver, Université PARIS 13