A ghost ring for the left-free double Burnside ring and an application to fusion systems

TL;DR

This paper introduces a ghost ring and mark homomorphism for the double Burnside ring of finite groups, simplifying its structure and applying it to fusion systems, with significant results on module classification and algebra semisimplicity.

Contribution

It defines a ghost ring and mark homomorphism for the double Burnside ring, extending the understanding of its structure and applications to fusion systems and module theory.

Findings

Simplified ring structure via ghost ring and mark homomorphism.

Determined simple modules and proved semisimplicity over characteristic zero.

Extended bijection between fusion systems and idempotents in the algebra.

Abstract

For a finite group $G$, we define a ghost ring and a mark homomorphism for the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and after tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two $p$-permutation bimodules $M$ and $N$ in terms of Brauer constructions of the bimodules $M$ and $N$. Over a field of characteristic 0 we determine the simple modules of the left-free double Burnside algebra and prove semisimplicity results for the bifree double Burnside algebra. These results carry over to results about biset-functor categories. Finally, we apply the ghost ring and mark homomorphism to fusion systems on a finite $p$-group. We extend a remarkable bijection, due to Ragnarsson and Stancu, between saturated fusion systems and certain idempotents of the bifree double Burnside algebra over $\ZZ_{(p)}$, to a bijection between all fusion systems and a larger set of idempotents in the bifree double Burnside algebra over $\QQ.$