Trihedral Soergel bimodules

TL;DR

This paper introduces trihedral Soergel bimodules and Hecke algebras, extending the dihedral case to a tricolored setting, and explores their connections to quantum groups and Dynkin diagrams.

Contribution

It generalizes dihedral Soergel bimodules to trihedral ones, establishing new Kazhdan-Lusztig combinatorics and classifying simple transitive 2-representations.

Findings

Introduction of trihedral Hecke algebras and Soergel bimodules

Establishment of their Kazhdan-Lusztig combinatorics

Classification of simple transitive 2-representations

Abstract

The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak{sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf{ADE}$ Dynkin diagrams. Using the quantum Satake correspondence between affine $\mathsf{A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak{sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $\mathsf{ADE}$ Dynkin diagrams.