Web Calculus and Tilting Modules in Type $C_2$

TL;DR

This paper develops a web calculus for type C2 and constructs a basis for morphisms between tensor products of fundamental representations, establishing an equivalence with tilting modules in quantum groups.

Contribution

It introduces a light leaves algorithm for C2 webs and proves an equivalence between the web category and tilting modules for quantum sp4, with minimal dependence on the base field.

Findings

Constructed a basis of morphisms using a light leaves algorithm.

Proved the equivalence of the web category with tilting modules for quantum sp4.

Established results hold when [2]_q ≠ 0.

Abstract

Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $\mathfrak{so}_5\cong \mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_q\ne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $\mathcal{U}_q^{\mathbb{Z}}(\mathfrak{sp}_4)$.