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