On character tables for fusion systems

Abstract

A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the determinant of $X\overline{X}$ (the square of the volume of this lattice) with the product of the orders of $S$-centralisers of fully $\mathcal{F}$-centralised $\mathcal{F}$-class representatives. This statement is exactly column orthogonality for the character table of $S$ when $\mathcal{F}={\mathcal{F}}_S(S)$. We prove the conjecture when $\mathcal{F}={\mathcal{F}}_S(G)$ is realised by some finite group $G$ with Sylow $p$-subgroup $S$, and for all simple fusion systems when $|S|\le p^4$. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of $\mathcal{F}$.

13 pages, 7 tables