Two-color Soergel Calculus and Simple Transitive 2-representations

Marco Mackaaij,Daniel Tubbenhauer

doi:10.4153/cjm-2017-061-2

Abstract

AbstractIn 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.

Highlights

Our construction gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs
The problem is that their classification is very hard and not well understood in general – except when certain specific conditions are satisfied, as for Soergel bimodules in type A [MM16b, Theorem 21] for example
The authors of [KMMZ16] studied the so-called small quotient of Soergel bimodules and their simple transitive 2-representations, for all finite Coxeter types. These 2-representations are given by categories on which the bimodules act by endofunctors and the bimodule maps by natural transformations

Summary

Introduction

Our construction gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs. Given a Dynkin diagram of type A, D or E with a bipartition, we define two degree-preserving, self-adjoint endofunctors Θs and Θt on the module category over the corresponding quiver – which is a zigzag algebra of type ADE – and, for each generating diagram in the two-color Soergel calculus, a natural transformation between composites of them such that all diagrammatic relations are preserved.

Results

Conclusion