Journal of Combinatorial Algebra

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Volume 1, Issue 1, 2017, pp. 63–125
DOI: 10.4171/JCA/1-1-4

Published online: 2016-12-13

Quantum Satake in type $A$. Part I

Ben Elias[1]

(1) University of Oregon, Eugene, USA

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 $n \geq 4$.

Keywords: Geometric Satake, quantum, Soergel bimodules, diagrammatics, webs

Elias Ben: Quantum Satake in type $A$. Part I. J. Comb. Algebra 1 (2017), 63-125. doi: 10.4171/JCA/1-1-4