## Skew cellularity of the Hecke algebras of type $G(\ell ,p,n)$

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- by Jun Hu, Andrew Mathas and Salim Rostam;
- Represent. Theory
**27**(2023), 508-573 - DOI: https://doi.org/10.1090/ert/646
- Published electronically: July 20, 2023
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## Abstract:

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras.

As an application of this general theory, the main result of this paper proves that the Hecke algebras of type $G(\ell ,p,n)$ are graded skew cellular algebras. In the special case when $p = 2$ this implies that the Hecke algebras of type $G(\ell ,2,n)$ are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting.

As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type $G(\ell ,p,n)$ are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

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## Bibliographic Information

**Jun Hu**- Affiliation: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
- MR Author ID: 635795
- Email: junhu404@bit.edu.cn
**Andrew Mathas**- Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
- MR Author ID: 349260
- ORCID: 0000-0001-7565-5798
- Email: andrew.mathas@sydney.edu.au
**Salim Rostam**- Affiliation: Univ. Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
- MR Author ID: 1204275
- ORCID: 0000-0001-5928-3920
- Email: salim.rostam@ens-rennes.fr
- Received by editor(s): December 14, 2022
- Received by editor(s) in revised form: February 6, 2023, and February 19, 2023
- Published electronically: July 20, 2023
- Additional Notes: The first author was supported by the National Natural Science Foundation of China (No. 12171029). The second author was supported, in part, by the Australian Research Council.
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory
**27**(2023), 508-573 - MSC (2020): Primary 20C08, 16G30, 05E10
- DOI: https://doi.org/10.1090/ert/646
- MathSciNet review: 4618071