2024/09/04 by Lawrence, Thomas · 1 citation
Mathematics · #20C20 (Secondary) #20C99 (Primary) 20D20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2409.03007
openalex publication_date 2024/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a fusion system over a p-group S. We study the complex character ring Rℂ(F) of F by applying techniques from modular character theory to F-stable characters. We use these techniques to investigate a conjecture posed by Jason Semeraro concerning the volume of Rℂ(F) as a ℤ-lattice. Proving it holds for all saturated fusion systems would allow for easy verification that a given set of linearly independent F-stable characters forms a ℤ-basis of Rℂ(F). We prove that this conjecture holds for all non-exotic fusion systems and a weakened conjecture holds for all fusion systems. We also show that any minimal counter example must be indecomposable by describing the characters of a product of two fusion systems. As a byproduct of our proof method, we describe the modular character rings of F, provide analogues of the decomposition and Cartan matrices for F-stable characters, and give a method for decomposing the regular character of S into F-stable constituents.