Schurian-finiteness of blocks of type A Hecke algebras

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.