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arXiv · 2208.07319

Categorification of integral group rings extended by one dimension

Andrew Schopieray
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Abstract

The integral group rings $\mathbb{Z}G$ 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.

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Andrew Schopieray. 2022-08-15. Categorification of integral group rings extended by one dimension. https://arxiv.org/abs/2208.07319

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