An asymptotic cell category for cyclic groups

TL;DR

This paper constructs braided pivotal monoidal categories linked to cyclic groups, extending Lusztig's theory of unipotent characters and fusion algebras to non-real reflection groups, including super modular categories.

Contribution

It introduces the first braided pivotal monoidal categories associated with non-real reflection groups, specifically cyclic groups, connecting to Malle's fusion algebras.

Findings

Constructed braided pivotal monoidal categories for cyclic groups

Extended Lusztig's unipotent character theory to non-real reflection groups

Linked fusion algebras to super modular categories

Abstract

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Brou\'e,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.