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

Abstract

For any algebra over an algebraically closed field , we say that an ‐module is Schurian if . We say that is Schurian‐finite if there are only finitely many isomorphism classes of Schurian ‐modules, and Schurian‐infinite otherwise. By work of Demonet, Iyama and Jasso, it is known that Schurian‐finiteness is equivalent to ‐tilting‐finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type Hecke algebras with quantum characteristic , 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 Hecke algebras (when ) 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.