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Trihedral Soergel bimodules

Volume 248 / 2020

Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer Fundamenta Mathematicae 248 (2020), 219-300 MSC: Primary 20C08; Secondary 17B10, 18D05, 18D10, 20F55. DOI: 10.4064/fm566-3-2019 Published online: 19 September 2019


The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum $\mathfrak {sl}_2$ representation category. It also establishes a precise relation between the simple transitive $2$-representations of both monoidal categories, which are indexed by bicolored $\mathsf {ADE}$ Dynkin diagrams.

Using the quantum Satake correspondence between affine $\mathsf {A}_{2}$ Soergel bimodules and the semisimple quotient of the quantum $\mathfrak {sl}_3$ representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan–Lusztig combinatorics, simple transitive $2$-representations corresponding to tricolored generalized $\mathsf {ADE}$ Dynkin diagrams.


  • Marco MackaayCenter for Mathematical Analysis, Geometry,
    and Dynamical Systems
    Departamento de Matemática
    Instituto Superior Técnico
    1049-001 Lisboa, Portugal
    Departamento de Matemática, FCT
    Universidade do Algarve
    Campus de Gambelas
    8005-139 Faro, Portugal
  • Volodymyr MazorchukDepartment of Mathematics
    Uppsala University
    Box 480
    SE-75106, Uppsala, Sweden
  • Vanessa MiemietzSchool of Mathematics
    University of East Anglia
    Norwich NR4 7TJ, United Kingdom
  • Daniel TubbenhauerInstitut für Mathematik
    Universität Zürich
    Winterthurerstrasse 190
    Campus Irchel, Office Y27J32
    CH-8057 Zürich, Switzerland

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