Representation Rings of Fusion Systems and Brauer Characters

Abstract

Let $\mathcal{F}$ be a fusion system over a $p$-group $S$. We study the complex character ring $R_{\mathbb{C}}(\mathcal{F})$ of $\mathcal{F}$ by applying techniques from modular character theory to $\mathcal{F}$-stable characters. We use these techniques to investigate a conjecture posed by Jason Semeraro concerning the volume of $R_{\mathbb{C}}(\mathcal{F})$ as a $\mathbb{Z}$-lattice. Proving it holds for all saturated fusion systems would allow for easy verification that a given set of linearly independent $\mathcal{F}$-stable characters forms a $\mathbb{Z}$-basis of $R_{\mathbb{C}}(\mathcal{F})$. We prove that this conjecture holds for all non-exotic fusion systems and a weakened conjecture holds for all fusion systems. We also show that any minimal counter example must be indecomposable by describing the characters of a product of two fusion systems. As a byproduct of our proof method, we describe the modular character rings of $\mathcal{F}$, provide analogues of the decomposition and Cartan matrices for $\mathcal{F}$-stable characters, and give a method for decomposing the regular character of $S$ into $\mathcal{F}$-stable constituents.

17 pages. Added Examples