The two-color Soergel calculus

TL;DR

This paper introduces a diagrammatic framework for Soergel bimodules in dihedral groups, connecting Temperley-Lieb categories and quantum Satake equivalences, with implications for representation theory.

Contribution

It provides a new diagrammatic presentation of Soergel bimodules for dihedral groups, integrating Temperley-Lieb categories and analyzing quantum parameters.

Findings

Diagrammatic presentation of Soergel bimodules

Embedding of Temperley-Lieb category as degree 0 morphisms

Quantum Satake equivalence for sl(2) at roots of unity

Abstract

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl(2). When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W .