On the irreducible Specht modules for Iwahori--Hecke algebras of type A with $q=-1$

TL;DR

This paper characterizes the finite set of partitions for which the Specht modules of the Iwahori--Hecke algebra at q=-1 are irreducible, focusing on partitions with both $lam$ and $lam'$ being 2-singular.

Contribution

It proves the finiteness of partitions with both $lam$ and $lam'$ 2-singular for which the Specht module is irreducible at q=-1.

Findings

Finitely many such partitions exist.

Characterization of irreducible Specht modules at q=-1.

Conditions on 2-singularity of partitions and their conjugates.

Abstract

Let $p$ be a prime and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{H}_n$ denote the Iwahori--Hecke algebra of the symmetric group $\mathfrak{S}_n$ over $\mathbb{F}$ at $q=-1$. We prove that there are only finitely many partitions $\lambda$ such that both $\lambda$ and $\lambda'$ are 2-singular and the Specht module $S^\lambda$ for $\mathcal{H}_{|\la|}$ is irreducible.