turkmath.org

Starting from the definition of a Burnside ring of a finite group, I will explain how Burnside ring of a fusion system can be defined. For a saturated fusion system $F$ defined on a $p$-group $S$, the Burnside ring $A(F)$ is defined as the Grothendieck ring of the monoid of $F$-stable $S$-sets. This monoid is a free commutative monoid with a canonical basis given by S. P. Reeh. In a recent work, joint with M. Gelvin and S. P. Reeh, we give an explicit formula that describes these basis elements as $S$-sets. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Moebius functions. I will explain how this formula is obtained.