Transfer and characteristic idempotents for saturated fusion systems

TL;DR

This paper develops a transfer map for p-local Burnside rings in saturated fusion systems, explicitly describes the characteristic idempotent, and explores its algebraic properties, leading to new insights and counterexamples in fusion system theory.

Contribution

It introduces a well-behaved transfer map for p-local Burnside rings and provides explicit descriptions of the characteristic idempotent in saturated fusion systems.

Findings

Constructed a transfer map from p-local Burnside ring of S to that of F.

Explicit description of the characteristic idempotent in terms of fixed points and transitive bisets.

Disproved a conjecture on the composition product of fusion systems.

Abstract

We construct a well-behaved transfer map from the p-local Burnside ring of the underlying p-group S to the p-local Burnside ring of a saturated fusion system F. Using this transfer map, we give new results on 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 explicitly both in terms of fixed points and as a linear combination of transitive bisets. Additionally, using fixed points we determine the map for Burnside rings given by multiplication with the characteristic idempotent, and show that this is the transfer map previously constructed. Applying these results, we show that for every saturated fusion system the ring generated by all (not necessarily idempotent) characteristic elements in the p-local double Burnside ring is isomorphic as rings to the p-local "single" Burnside ring of the fusion system, and we disprove a conjecture by Park-Ragnarsson-Stancu on the composition product of fusion systems.