Fusion rings for quantum groups

TL;DR

This paper investigates the structure of fusion rings associated with quantum groups at roots of unity, providing combinatorial descriptions, presentations, and explicit computations across various types including G2.

Contribution

It offers a unified description of fusion rings for classical types and extends known results, including explicit calculations for G2.

Findings

Identification of fusion rings with combinatorial rings in type A

Description of sp(2n)-fusion ring via noncommutative symmetric functions

Presentation of fusion rings as quotients of polynomial rings

Abstract

We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from [KS] and give a similar description of the sp(2n)-fusion ring in terms of noncommutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings extending known results in special cases. Finally we also compute the fusion rings for type G2.