Representation Rings of Fusion Systems and Brauer Characters

TL;DR

This paper explores the structure of the ring of stable characters in fusion systems, establishing new connections to modular characters and deriving formulas for character tables, with implications for understanding fusion system representations.

Contribution

It introduces a novel link between fusion system stable characters and modular characters of finite groups, enabling new calculations and structural insights.

Findings

Determined the rank of the $$-stable character ring over fields of positive characteristic.

Derived a formula for the determinant of the $$-character table.

Proved the squared value of the determinant is a power of $p$ for all saturated fusion systems.

Abstract

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.