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On character tables for fusion systems

Thomas Lawrence, Jason Semeraro

TL;DR

This work extends a conjectural relation between the $p$-part of the determinant of a fusion-system character-table and centraliser data to saturated fusion systems. It develops the ring of $\mathcal{F}$-stable virtual characters, defines indecomposable constituents, and proves a key lattice-technical lemma enabling transfer of bases from subsystems to the whole system. The authors establish the conjecture for group-realizable fusion systems and for all simple fusion systems on $p$-groups of order at most $p^4$, applying detailed case analyses of exotic simple systems and, where needed, computer-assisted checks for small primes. The results connect column orthogonality in fusion-system character tables with centraliser structure, and they illuminate the role of saturation and principal blocks in ensuring the conjectural identity holds. Overall, the paper advances understanding of how fusion-system representation theory mirrors classical group representation theory, with explicit classifications and constructive bases in low-order cases.

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 virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a conjecture of the second author which equates the $p$-part of $|$det$(X)|^2$ 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 $|S| \le p^4$.

On character tables for fusion systems

TL;DR

This work extends a conjectural relation between the -part of the determinant of a fusion-system character-table and centraliser data to saturated fusion systems. It develops the ring of -stable virtual characters, defines indecomposable constituents, and proves a key lattice-technical lemma enabling transfer of bases from subsystems to the whole system. The authors establish the conjecture for group-realizable fusion systems and for all simple fusion systems on -groups of order at most , applying detailed case analyses of exotic simple systems and, where needed, computer-assisted checks for small primes. The results connect column orthogonality in fusion-system character tables with centraliser structure, and they illuminate the role of saturation and principal blocks in ensuring the conjectural identity holds. Overall, the paper advances understanding of how fusion-system representation theory mirrors classical group representation theory, with explicit classifications and constructive bases in low-order cases.

Abstract

A character table for a saturated fusion system on a finite -group is the square matrix of values associated to a basis of virtual -stable ordinary characters of . We investigate a conjecture of the second author which equates the -part of det with the product of the orders of -centralisers of fully -centralised -class representatives. This statement is exactly column orthogonality for the character table of when . We prove the conjecture when is realised by some finite group with Sylow -subgroup , and for all simple fusion systems when .

Paper Structure

This paper contains 10 sections, 21 theorems, 49 equations, 7 tables.

Table of Contents

  1. Introduction
  2. The ring of $\mathcal{F}$-stable virtual characters
  3. Proof of Theorem \ref{['t:groupcase']}
  4. Characters in the principal block
  5. Simple exotic fusion systems on $p$-groups of order $p^4$
  6. Proof of Theorem \ref{['t:main']}
  7. The case $\mathcal{F}=\mathcal{G}$
  8. The case $\mathcal{F}=\mathcal{F}_{(1)}$
  9. The case $\mathcal{F}=O^{p'}(\mathcal{F}_{(1)})$
  10. The case $p \le 5$

Key Result

Theorem 1.2

Conjecture c:main holds when $\mathcal{F}=\mathcal{F}_S(G)$ is the fusion system of a finite group $G$ in which $S$ is a Sylow $p$-subgroup.

Theorems & Definitions (44)

  • Conjecture 1.1: Semeraro
  • Theorem 1.2: Lawrence--Olsson
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 34 more