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A Soergel-like category for complex reflection groups of rank one

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Abstract

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by generators and relations. This ring turns out to be an extension of the Hecke algebra of the reflection group W and a free module of rank \(|W| (|W|-1)+1\) over the base ring. We also show that it is a generically semisimple algebra if defined over the complex numbers.

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Acknowledgements

We thank Pierre-Emmanuel Chaput, Eirini Chavli, Anthony Henderson, Ivan Marin and Ulrich Thiel for useful discussions. We acknowledge the support of Universität Stuttgart during the early stages of this project. We are grateful to the anonymous referee for the careful reading and the insightful suggestions. The first author was funded by an ARC Grant (Grant number DP170101579). The second author was funded by the Grant RIN–ARTIQ of the Région Normandie.

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Authors and Affiliations

  1. School of Mathematics and Statistics F07, The University of Sydney, NSW, 2006, Australia

    Thomas Gobet

  2. Normandie Univ, Unicaen, CNRS, LMNO, 14000, Caen, France

    Anne-Laure Thiel

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  1. Thomas Gobet
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  2. Anne-Laure Thiel
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Correspondence to Thomas Gobet.

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Gobet, T., Thiel, AL. A Soergel-like category for complex reflection groups of rank one. Math. Z. 295, 643–665 (2020). https://doi.org/10.1007/s00209-019-02358-x

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