On webs in quantum type $C$

Abstract

We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}({\mathfrak{sp}}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{FundRep}(U_q({\mathfrak{sp}}_6))$ of quantum ${\mathfrak{sp}}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\mathbf{Web}({\mathfrak{sp}}_6)\to \mathbf{FundRep}(U_q({\mathfrak{sp}}_6))$, that all $\mathrm{Hom}$-spaces in $\mathbf{Web}({\mathfrak{sp}}_6)$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\mathbf{Web}({\mathfrak{sp}}_6)$ is $1$-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum ${\mathfrak{sp}}_6$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

arXiv comment34 pages, many color figures

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