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 M\"obius functions.