On webs in quantum type C

Abstract

We study webs in quantum type , focusing on the rank three case. We define a linear pivotal category Web(6) diagrammatically by generators and relations, and conjecture that it is equivalent to the category FundRep((6)) of quantum 6 representations generated by the fundamental representations, for generic values of the parameter. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor Web(6) → FundRep((6)), that all Hom-spaces in Web(6) are finite-dimensional, and that the endomorphism algebra of the monoidal unit in Web(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 6 link invariants, akin to the Kauffman bracket description of the Jones polynomial.

Web calculus and tilting modules in type $C_2$

“…Proving that the diagrammatic category was equivalent to Fund.sl n / turned out to be difficult, but was eventually carried out by Cautis, Kamnitzer, and Morrison using skew Howe duality [6]. Recently, a conjectural description of sp 6 -webs has appeared in a preprint by Rose and Tatham [26].…”