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

TL;DR

This paper proves that blocks of type A Hecke algebras with weight at least 2 in quantum characteristic e ≥ 3 are Schurian-infinite, linking Schurian-finiteness to representation-finiteness.

Contribution

It establishes the Schurian-infinite nature of certain blocks of type A Hecke algebras, extending previous work and clarifying conditions for Schurian-finiteness.

Findings

Blocks of weight ≥ 2 in type A Hecke algebras are Schurian-infinite for e ≥ 3.

Schurian-finiteness coincides with representation-finiteness in these algebras.

The result builds on prior research by Ariki and others.

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. In this paper, we build on the work of Ariki and the second author to show that all blocks of type $A$ Hecke algebras of weight at least $2$ in quantum characteristic $e \geq 3$ are Schurian-infinite. This proves that if $e \geq 3$ then blocks of type $A$ Hecke algebras are Schurian-finite if and only if they are representation-finite.