Schurian-finiteness of blocks of type A Hecke algebras

Ariki, Susumu, Lyle, Sinéad ORCID: https://orcid.org/0000-0002-6032-7721 and Speyer, Liron (2023) Schurian-finiteness of blocks of type A Hecke algebras. Journal of the London Mathematical Society, 108 (6). pp. 2333-2376. ISSN 0024-6107

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Abstract

For any algebra (Formula presented.) over an algebraically closed field (Formula presented.), we say that an (Formula presented.) -module (Formula presented.) is Schurian if (Formula presented.). We say that (Formula presented.) is Schurian-finite if there are only finitely many isomorphism classes of Schurian (Formula presented.) -modules, and Schurian-infinite otherwise. By work of Demonet, Iyama and Jasso, it is known that Schurian-finiteness is equivalent to (Formula presented.) -tilting-finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type (Formula presented.) Hecke algebras with quantum characteristic (Formula presented.), all blocks of weight at least 2 are Schurian-infinite in any characteristic. Weight 0 and 1 blocks are known by results of Erdmann and Nakano to be representation finite, and are therefore Schurian-finite. This means that blocks of type (Formula presented.) Hecke algebras (when (Formula presented.)) are Schurian-infinite if and only if they have wild representation type if and only if the module category has finitely many wide subcategories. Along the way, we also prove a graded version of the Scopes equivalence, which is likely to be of independent interest.

Item Type: Article
Additional Information: Research Funding: Japan Society for the Promotion of Science. Grant Numbers: 18K03212, 21K03163, 20K22316; London Mathematical Society. Grant Number: ENF20-21-02
Faculty \ School: Faculty of Science > School of Mathematics
UEA Research Groups: Faculty of Science > Research Groups > Algebra and Combinatorics
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Depositing User: LivePure Connector
Date Deposited: 06 Jan 2024 01:41
Last Modified: 11 Jan 2024 01:39
URI: https://ueaeprints.uea.ac.uk/id/eprint/94076
DOI: 10.1112/jlms.12808

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