Quantum Satake in type $A$. Part I

Abstract

We give an interpretation of ${\mathfrak{sl}}_n$-webs as morphisms between certain singular Soergel bimodules. We explain how this is a combinatorial, algebraic version of the geometric Satake equivalence (in type $A$). We then $q$-deform the construction, giving an equivalence between representations of $U_q({\mathfrak{sl}}_n)$ and certain singular Soergel bimodules for a $q$-deformed Cartan matrix.

In this paper, we discuss the general case but prove only the case $n=2,3$. In the sequel we will prove $n\ge 4$.