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 prove a conjecture of the second author which equates the determinant of $X{\overline{X}}^T$ (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 first give an elementary proof when $\mathcal{F}={\mathcal{F}}_S(G)$ is realised by a finite group $G$ with Sylow $p$-subgroup $S$, and then treat the general case by exploiting the characteristic idempotent ${\omega }_{\mathcal{F}}$ of $\mathcal{F}$ in the $p$-local double Burnside ring. The two ingredients are that ${\omega }_{\mathcal{F}}$ acts on class functions as the operator of averaging over $\mathcal{F}$-classes, which forces the lattice of $\mathcal{F}$-stable characters to be self-dual at $p$, and that this averaging operator preserves $\ZZ_{(p)}$-valued class functions, which forces $|x^{\mathcal{F}}{|}_p=[S:C_S(x)]$ for every fully $\mathcal{F}$-centralised $x\in S$. The latter equality was conjectured in an earlier version of this paper.

arXiv comment16 pages, 1 table