Representation Rings of Fusion Systems and Brauer Characters

Let $\mathcal{F}$ be a saturated fusion system on a $p$-group $S$. We study the ring $R(\mathcal{F})$ of $\mathcal{F}$-stable characters by exploiting a new connection to the modular characters of a finite group $G$ with $\mathcal{F} = \mathcal{F}_S(G)$. We utilise this connection to find the rank of the $\mathcal{F}$-stable character ring over fields with positive characteristic. We use this theory to derive a decomposition of the regular representation for a fixed basis $B$ of the ring of complex $\mathcal{F}$-stable characters and give a formula for the absolute value of the determinant of the $\mathcal{F}$-character table with respect to $B$ (the matrix of the values taken by elements of $B$ on each $\mathcal{F}$-conjugacy class) for a wide class of saturated fusion systems, including all non-exotic fusion systems, and prove this value squared is a power of $p$ for all saturated fusion systems.