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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Simple transitive $ 2$-representations of small quotients of Soergel bimodules


Authors: Tobias Kildetoft, Marco Mackaay, Volodymyr Mazorchuk and Jakob Zimmermann
Journal: Trans. Amer. Math. Soc. 371 (2019), 5551-5590
MSC (2010): Primary 18D05; Secondary 16G10, 17B10
DOI: https://doi.org/10.1090/tran/7456
Published electronically: August 8, 2018
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Abstract: In all finite Coxeter types but $ I_2(12)$, $ I_2(18)$, and $ I_2(30)$, we classify simple transitive $ 2$-representations for the quotient of the $ 2$-category of Soergel bimodules over the coinvariant algebra which is associated with the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $ 2$-representations are exhausted by cell $ 2$-representations. However, in Coxeter types $ I_2(2k)$, where $ k\geq 3$, there exist simple transitive $ 2$-representations which are not equivalent to cell $ 2$-representations.


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Additional Information

Tobias Kildetoft
Affiliation: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden
Address at time of publication: Department of Mathematics, Århus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
Email: kildetoft@math.au.dk

Marco Mackaay
Affiliation: Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal; and Departamento de Matemática, FCT, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal
Email: mmackaay@ualg.pt

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden
Email: mazor@math.uu.se

Jakob Zimmermann
Affiliation: Department of Mathematics, Uppsala University, Box. 480, SE-75106, Uppsala, Sweden
Email: jakob.zimmermann@math.uu.se

DOI: https://doi.org/10.1090/tran/7456
Received by editor(s): May 12, 2016
Received by editor(s) in revised form: October 20, 2017
Published electronically: August 8, 2018
Additional Notes: The first author was partially supported by Knut and Alice Wallenbergs Foundation
The third author was partially supported by the Swedish Research Council, Knut and Alice Wallenbergs Foundation, and Göran Gustafsson Foundation.
Article copyright: © Copyright 2018 American Mathematical Society