Two-color Soergel calculus and simple transitive 2-representations

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.

37 pages, many colored figures, revised version, comments welcome, to appear in Canad. J. Math

References in corpus (4)

• The two-color Soergel calculus
• Diagram categories for ${\text{U}}_q$-tilting modules at roots of unity
• Quantum Satake in type A: part I
• Trihedral Soergel bimodules

Cited by in corpus (7)

• Simple transitive 2-representations via (co)algebra 1-morphisms
• Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types
• Bimodules over uniformly oriented A_n quivers with radical square zero
• Monoidal categories, representation gap and cryptography
• Asymptotics in finite monoidal categories
• Algebraic properties of zigzag algebras
• $A_{\infty }$-deformations of zigzag algebras via Ginzburg dg algebras