Cellularity of endomorphism algebras of tilting objects

Bellamy, G. and Thiel, U. (2022) Cellularity of endomorphism algebras of tilting objects. Advances in Mathematics, 404(Part A), 108387. (doi: 10.1016/j.aim.2022.108387)

Abstract

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer [4]. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases—and thus a general theory of “cell modules”—for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category O of the rational Cherednik algebra. For real reflection groups these bases are cellular.

• Item Type: Articles
• Additional Information: The first author was partially supported by EPSRC grant EP/N005058/1. The second author was partially supported by the DFG SPP 1489, by a Research Support Fund from the Edinburgh Mathematical Society, and by the Australian Research Council Discovery Projects grant no. DP160103897.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Bellamy, Professor Gwyn
• Authors: Bellamy, G., and Thiel, U.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Advances in Mathematics
• Publisher: Elsevier
• ISSN: 0001-8708
• ISSN (Online): 1090-2082
• Published Online: 26 April 2022
• Copyright Holders: Copyright © 2022 Elsevier Inc.
• First Published: First published in Advances in Mathematics 404(Part A): 108387
• Publisher Policy: Reproduced in accordance with the publisher copyright policy

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
171924		Symplectic representation theory	Gwyn Bellamy	Engineering and Physical Sciences Research Council (EPSRC)	EP/N005058/1	M&S - Mathematics