Web calculus and tilting modules in type $C_2$

Abstract

Using Kuperberg’s web calculus (1996), 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 ${\mathfrak{sp}}_4$ (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 $U_q^{\mathcal{A}}({\mathfrak{sp}}_4)$.