Simple transitive $2$-representations of Soergel bimodules in type $B_2$

TL;DR

This paper classifies simple transitive 2-representations of Soergel bimodules in type B2, showing they are all equivalent to cell 2-representations, and explores properties for dihedral groups.

Contribution

It proves that all simple transitive 2-representations in type B2 are equivalent to cell 2-representations, providing a complete classification in this case.

Findings

All simple transitive 2-representations in type B2 are cell 2-representations.

Describes properties of Soergel bimodules for dihedral groups.

Provides groundwork for understanding 2-representations in finite Coxeter types.

Abstract

We prove that every simple transitive $2$-representation of the fiat $2$-category of Soergel bimodules (over the coinvariant algebra) in type $B_2$ is equivalent to a cell $2$-representation. We also describe some general properties of the $2$-category of Soergel bimodules for arbitrary finite Dihedral groups.