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

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