TY - JOUR

T1 - Schurian-finiteness of blocks of type A Hecke algebras

AU - Ariki, Susumu

AU - Lyle, Sinéad

AU - Speyer, Liron

N1 - Research Funding: Japan Society for the Promotion of Science. Grant Numbers: 18K03212, 21K03163, 20K22316; London Mathematical Society. Grant Number: ENF20-21-02

PY - 2023/12

Y1 - 2023/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85168329005&partnerID=8YFLogxK

U2 - 10.1112/jlms.12808

DO - 10.1112/jlms.12808

M3 - Article

VL - 108

SP - 2333

EP - 2376

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 6

ER -