Sinéad Lyle | Searcharxiv

arXiv2025

Schurian-finiteness of blocks of type B Hecke algebras

Schurian-finiteness, also known as $τ$-tilting finiteness, is equivalent to the finiteness of various representation theoretic objects such as wide subcategories. The first three authors classified Schurian-finite blocks of type A Hecke algebras in [ALS23]. Here we study the Schurian-finiteness of blocks of type B Hecke algebras, and determine the Schurian-finiteness of all blocks if the Hecke algebra is `non-integral', and for almost all blocks in the integral case. The only remaining cases are a small number of blocks in defect $3$ when $e=3$, and a family of blocks in defects $3$ and $4$ for $e\geqslant4$. The classification is mostly achieved by methods using decomposition numbers, with many degenerate cases requiring direct study using standard methods from the representation theory of quivers.

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arXiv2023

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

For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. By work of Demonet, Iyama and Jasso it is known that Schurian-finiteness is equivalent to $τ$-tilting-finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type $A$ Hecke algebras with quantum characteristic $e\geq 3$, 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 $A$ Hecke algebras (when $e\geq 3$) 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.

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arXiv2023

Schurian-finiteness of blocks of type $A$ Hecke algebras II

For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. In this paper, we build on the work of Ariki and the second author to show that all blocks of type $A$ Hecke algebras of weight at least $2$ in quantum characteristic $e \geq 3$ are Schurian-infinite. This proves that if $e \geq 3$ then blocks of type $A$ Hecke algebras are Schurian-finite if and only if they are representation-finite.

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