May, S orcid.org/0000-0002-1870-3690 (2023) Non-Commutative Resolutions for the Discriminant of the Complex Reflection Group G(m, p, 2). Algebras and Representation Theory, 26. pp. 2841-2876. ISSN 1386-923X
Abstract
We show that for the family of complex reflection groups G = G(m, p,2) appearing in the Shephard–Todd classification, the endomorphism ring of the reduced hyperplane arrangement A(G) is a non-commutative resolution for the coordinate ring of the discriminant Δ of G. This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for Δ from A(G) and decompose it using data from the irreducible representations of G. For G(m, p,2) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen–Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement A(G) will be a non-commutative resolution. For the groups G(m,1,2), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.
Metadata
Item Type: | Article |
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Authors/Creators: | |
Copyright, Publisher and Additional Information: | © The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | Complex reflection groups; Hyperplane arrangements; Cohen-Macaulay modules; Matrix factorizations; Noncommutative desingularization |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 23 Jan 2023 16:09 |
Last Modified: | 18 Jan 2024 15:35 |
Published Version: | http://dx.doi.org/10.1007/s10468-022-10193-8 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s10468-022-10193-8 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:195537 |