Schurian-finiteness of blocks of type B Hecke algebras

TL;DR

This paper investigates the Schurian-finiteness of blocks in type B Hecke algebras, providing a near-complete classification for both integral and non-integral cases, with some remaining unresolved cases.

Contribution

It extends the classification of Schurian-finite blocks from type A to type B Hecke algebras, covering most cases and employing decomposition numbers and quiver methods.

Findings

Classified Schurian-finiteness for non-integral blocks.

Determined Schurian-finiteness for almost all integral blocks.

Identified remaining cases in defect 3 and 4 for further study.

Abstract

Schurian-finiteness, also known as $\tau$-tilting finiteness, is equivalent to the finiteness of various representation theoretic objects such as wide subcategories. The first three authors classified Schurian-finite blocks of type A Hecke algebras in [ALS23]. Here we study the Schurian-finiteness of blocks of type B Hecke algebras, and determine the Schurian-finiteness of all blocks if the Hecke algebra is `non-integral', and for almost all blocks in the integral case. The only remaining cases are a small number of blocks in defect $3$ when $e=3$, and a family of blocks in defects $3$ and $4$ for $e\geqslant4$. The classification is mostly achieved by methods using decomposition numbers, with many degenerate cases requiring direct study using standard methods from the representation theory of quivers.