Two-color Soergel calculus and simple transitive 2-representations

Abstract

In this paper we complete the 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 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.