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# Two-color Soergel calculus and simple transitive 2-representations

• Marco Mackaaij,
Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
• Daniel Tubbenhauer,
Mathematisches Institut, Universität Bonn , Endenicher Allee 60, D-53115 Bonn, Germany
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## 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.
 Keywords: $2$-representation theory, categorification, Soergel bimodule, Kazhdan--Lusztig theory, Hecke algebras for dihedral groups, zigzag algebra
 MSC Classifications: 20C08 - Hecke algebras and their representations 17B10 - Representations, algebraic theory (weights) 18D05 - Double categories, $2$-categories, bicategories and generalizations 18D10 - Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]

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