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The ABC of p-cells

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Abstract

Parallel to the very rich theory of Kazhdan–Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson in Categorification and higher representation theory, American Mathematical Society, Providence, 2017). Our main technical tools are the star-operations introduced by Kazhdan–Lusztig (Invent. Math. 53(2):65–184, 1979) which have interesting numerical consequences for the p-canonical basis. As an application, we explicitly describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson–Schensted correspondence. Moreover, we show that Kazhdan–Lusztig cells in finite types B and C decompose into p-cells for \(p > 2\).

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Acknowledgements

Since this paper is part of the author’s PhD-thesis, I would like to thank my supervisor Geordie Williamson—his constant support, his inspiring mathematical vision and his enthusiasm were crucial for the success of my PhD project. Moreover, I am grateful to the Max-Planck Institute for Mathematics for the perfect working conditions and the financial support for my research visit in Sydney. The project was started at the School of Mathematics and Statistics at the University of Sydney and finished at the Max-Planck Institute for Mathematics in Bonn. I would also like to thank Simon Riche for detailed comments and Monty McGovern for providing a preliminary version of [18] and of his joint work in progress with Thomas Pietraho.

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  1. Université Clermont Auvergne, CNRS, LMBP, F-63000, Clermont-Ferrand, France

    Lars Thorge Jensen

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Jensen, L.T. The ABC of p-cells. Sel. Math. New Ser. 26, 28 (2020). https://doi.org/10.1007/s00029-020-0552-1

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