Two-color Soergel calculus and simple transitive 2-representations

TL;DR

This paper classifies simple transitive 2-representations of Soergel bimodules in finite dihedral type, providing explicit constructions and criteria for equivalence, and extends some results to infinite dihedral type.

Contribution

It completes the ADE-like classification of simple transitive 2-representations in finite dihedral type and introduces explicit constructions using bipartite graphs and zigzag algebras.

Findings

Explicit graded constructions of 2-representations

Criteria for equivalence of 2-representations

Extension to infinite dihedral type

Abstract

In this paper we complete the $\mathrm{ADE}$-like classification of simple transitive $2$-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of $\mathrm{ADE}$ type to give an explicit construction of a graded (non-strict) version of all these $2$-representations. Moreover, we give simple combinatorial criteria for when two such $2$-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations. Finally, our construction also gives a large class of simple transitive $2$-representations in infinite dihedral type for general bipartite graphs.