The Grothendieck Ring of a Family of Spherical Categories

TL;DR

This paper computes explicit fusion rules and characters for a $q$-parameterized spherical category related to Young diagrams, advancing understanding of its Grothendieck ring structure.

Contribution

It provides closed-form formulas for fusion rules and characters of the spherical category, connecting combinatorics with category theory.

Findings

Fusion rules expressed via Littlewood-Richardson coefficients

Explicit character formulas including generating functions

Enhanced understanding of the Grothendieck ring structure

Abstract

The first author constructed a $q$-parameterized spherical category $\sC$ over $\mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.