Web Calculus and Tilting Modules in Type $C_2$

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 ${\mathcal{U}}_q^{\mathbb{Z}}({\mathfrak{sp}}_4)$.

arXiv comment40 pages, many figures and in color, second version with improved exposition