The Rank One property for free Frobenius extensions

Bellamy, G. and Thiel, U. (2023) The Rank One property for free Frobenius extensions. Comptes Rendus Mathématique, 361, pp. 1341-1348. (doi: 10.5802/crmath.502)

Abstract

A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over Q when restricted to each block of the algebra. In this paper, we show that if H is a prime algebra that is a free Frobenius extension over a regular central subalgebra R, and the centre of H is normal Gorenstein, then each central quotient A of H by a maximal ideal m of R satisfies the rank-one property with respect to the Cartan matrix of A. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras. In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

• Item Type: Articles
• Additional Information: The first author was partially supported by a Research Project Grant from the Leverhulme Trust and by the EPSRC grant EP-W013053-1. This work is a contribution to the SFB-TRR 195 ”Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG).
• 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: Comptes Rendus Mathématique
• Publisher: French Academy of Sciences
• ISSN: 1631-073X
• ISSN (Online): 1778-3569
• Copyright Holders: Copyright: © 2023 The Author(s)
• First Published: First published in Comptes Rendus Mathématique 361: 1341-1348
• Publisher Policy: Reproduced under a Creative Commons licence