Crystal rules for $(\ell,0)$-JM partitions

TL;DR

This paper provides a new combinatorial description of $( ell,0)$-JM partitions using crystal theory and hook length conditions, extending previous results to a broader class of partitions.

Contribution

It introduces a novel crystal-based characterization of $( ell,0)$-JM partitions, generalizing earlier interpretations and connecting them with hook length criteria.

Findings

Extended the classification of $( ell,0)$-JM partitions.

Connected crystal theory with hook length conditions.

Generalized previous results to all $( ell,0)$-JM partitions.

Abstract

Vazirani and the author \cite{BV} gave a new interpretation of what we called $\ell$-partitions, also known as $(\ell,0)$-Carter partitions. The primary interpretation of such a partition $\lambda$ is that it corresponds to a Specht module $S^{\lambda}$ which remains irreducible over the finite Hecke algebra $H_n(q)$ when $q$ is specialized to a primitive $\ell^{th}$ root of unity. To accomplish this we relied heavily on the description of such a partition in terms of its hook lengths, a condition provided by James and Mathas. In this paper, I use a new description of the crystal $reg_\ell$ which helps extend previous results to all $(\ell,0)$-JM partitions (similar to $(\ell,0)$-Carter partitions, but not necessarily $\ell$-regular), by using an analogous condition for hook lengths which was proven by work of Lyle and Fayers.