2022/08/15 by Andrew Schopieray, Schopieray, Andrew · 2 citations
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2208.07319
openalex publication_date 2022/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The integral group rings ℤG for finite groups G are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius-Perron dimensions take exactly one value distinct from 1. Our goal is a set of results which assist in characterizing when such fusion rings are categorifiable. As proof of concept, we completely classify the categorifiable near-group fusion rings for an infinite collection of finite abelian groups, a task that to-date has only been completed for three such groups.