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

TL;DR

This paper proves that most blocks of type $A$ Hecke algebras with quantum characteristic $e ext{≥}3$ are Schurian-infinite if they have weight at least 2, linking Schurian-finiteness to representation type.

Contribution

It establishes the Schurian-infinite property for blocks of type $A$ Hecke algebras with weight ≥ 2 and provides a graded version of the Scopes equivalence.

Findings

Blocks of weight ≥ 2 are Schurian-infinite in characteristic zero and positive characteristic.

Weight 0 and 1 blocks are Schurian-finite, aligning with their finite representation type.

Schurian-infinite blocks correspond to wild representation type and finitely many wide subcategories.

Abstract

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 $\tau$-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.