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

Susumu Ariki, Liron Speyer

Published 2021-12-21, updated 2022-03-28Version 2

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 if $e\geq 3$, then principal blocks of type $A$ Hecke algebras are Schurian-finite if and only if they have weight $0$ or $1$. By results of Erdmann and Nakano, this means that these principal blocks are Schurian-finite if and only if they have finite representation type under our assumption on $e$, or equivalently that they are Schurian-infinite if and only if they have wild representation type. Along the way, we also prove a graded version of the Scopes equivalence, which may be of independent interest. If $e=2$ and $p\ne 2$, then blocks of weight $0$, $1$ or $2$ are Schurian-finite. Again by Erdmann and Nakano, these weight $2$ blocks have infinite tame representation type. When $e=2$, blocks of weight at least $3$ have wild representation type, but our methods cannot determine whether these blocks are Schurian-infinite. Our methods do, however, apply to a large number of other blocks, and we discuss these in the final section.