Admissibility and the $C_2$ Spider

TL;DR

This paper identifies a multiplicity-free subcategory within the representation category of quantum Sp(4), providing dimension formulas for Hom spaces and introducing an admissibility concept for ribbon graph invariants.

Contribution

It constructs a multiplicity-free subcategory of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ and describes Hom space dimensions, extending understanding of quantum group representations and their graphical calculus.

Findings

A full subcategory with multiplicity-free Hom spaces is identified.

Dimension formulas for Hom spaces are provided, including at roots of unity.

An admissibility notion for ribbon graph invariants is introduced.

Abstract

A tensor category is multiplicity-free if for any objects $A,B,C$ we have that $\mathrm{Hom}(A\otimes B\otimes C,\mathbb{C})$ is either $0$ or $1$ dimensional. It is known that $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ is not multiplicty-free. We find a full subcategory of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ which is multiplicty-free. A description of the dimension of these $\mathrm{Hom}$ spaces is given for this subcategory, including when $q$ is a root of unity. The methods used arise from the description, given by Kuperberg, of $Rep^{uni}(U_q(\mathfrak{sp}(4)))$ as a spider. The main tool is the recursive definition of clasps given by Kim. In particular, we provide an appropriate notion of admissibility when looking at the $\mathrm{Sp}(4)_k$ ribbon graph invariants with restricted edge labels.