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Two-color Soergel Calculus and Simple Transitive 2-representations

Part of: Special aspects of infinite or finite groups Categories with structure Representation theory of groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 January 2019

Marco Mackaaij  and
Daniel Tubbenhauer
Marco Mackaaij
Affiliation:
Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal Email: [email protected]
Daniel Tubbenhauer
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasses 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, Switzerland Email: [email protected] URL: www.dttubbenhauer.com
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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 theorycategorificationSoergel bimoduleKazhdan–Lusztig theoryHecke algebras for dihedral groupszigzag algebra

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 17B10: Representations, algebraic theory (weights) 18D05: Double categories, $2$-categories, bicategories and generalizations 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20F55: Reflection and Coxeter groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1523 - 1566
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Author M. M. is partially supported by FCT/Portugal through the project UID/MAT/04459/2013

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