On the Basis of the Burnside Ring of a Fusion System

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 Möbius functions.

arXiv comment26 pages. Updated references [5] and [9]