On the Basis of the Burnside Ring of a Fusion System

Matthew Gelvin; Sune Precht Reeh; Ergun Yalcin

On the Basis of the Burnside Ring of a Fusion System

Algebraic Topology 2014-03-31 v2 Group Theory Representation Theory

Authors: Matthew Gelvin , Sune Precht Reeh , Ergun Yalcin

Abstract

We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$ -stable $S$ -sets for a saturated fusion system $\mathcal{F}$ defined on a $p$ -group $S$ . It is shown by S. P. Reeh that the monoid of $\mathcal{F}$ -stable sets is a free commutative monoid with canonical basis $\{\alpha_P\}$ . We give an explicit formula that describes $\alpha_P$ as an $S$ -set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified M\"obius functions.

Keywords

Burnside ring fusion system S-set broken chains Möbius function p-group canonical basis

Cite

@article{arxiv.1403.6053,
  title  = {On the Basis of the Burnside Ring of a Fusion System},
  author = {Matthew Gelvin and Sune Precht Reeh and Ergun Yalcin},
  journal= {arXiv preprint arXiv:1403.6053},
  year   = {2014}
}

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26 pages. Updated references [5] and [9]

On the Basis of the Burnside Ring of a Fusion System

Abstract

Keywords

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