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

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 $τ$-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.