On webs in quantum type C

• Published in: Canadian journal of mathematics Vol. 74; no. 3; pp. 793 - 832
• Main Authors: Rose, David E. V., Tatham, Logan C.
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical Society 01.06.2022; Cambridge University Press
• Subjects: Algebra; Generators; Mathematical analysis; Polynomials; Representations; Scalars; Webs; 57K16; 17B37; pivotal category; Webs; quantum groups; 18M30
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/S0008414X21000109

We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category $\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ 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 $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ , that all $\textrm {Hom}$ -spaces in $\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\textbf {Web}(\mathfrak {sp}_{6})$ is one-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.