On character tables for fusion systems

2025/10/10 by Thomas B. Lawrence, Lawrence, Thomas, Jason Semeraro +1
Mathematics · #19A22 #20C15 #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2510.09277

openalex publication_date 2025/10/10 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

A character table X for a saturated fusion system F on a finite p-group S is the square matrix of values associated to a basis of the lattice of virtual F-stable ordinary characters of S. We investigate a conjecture of the second author which equates the determinant of X X (the square of the volume of this lattice) with the product of the orders of S-centralisers of fully F-centralised F-class representatives. This statement is exactly column orthogonality for the character table of S when F=F_S(S). We prove the conjecture when F=F_S(G) is realised by some finite group G with Sylow p-subgroup S, and for all simple fusion systems when |S| ≤ p⁴. We also put forward a potential strategy for the general case, which would exploit properties of the characteristic idempotent of F.

Citations

• Representation Rings of Fusion Systems and Brauer Characters
• Fusion invariant characters of p-groups
• Fusion-invariant representations for symmetric groups
• Uniqueness of factorization for fusion-invariant representations
• Realizing fusion systems inside finite groups

Related

• Representation Rings of Fusion Systems and Brauer Characters
• Fusion invariant characters of p-groups
• Uniqueness of factorization for fusion-invariant representations
• On the Galois symmetries for the character table of an integral fusion\n category
• Fusion systems over a Sylow p-subgroup of G₂(p)