Specht modules labelled by hook bipartitions I

TL;DR

This paper studies graded Specht modules labeled by hook bipartitions in cyclotomic Khovanov--Lauda--Rouquier algebras, explicitly describing their structure and composition series for type B Hecke algebras with quantum characteristic at least three.

Contribution

It provides an explicit description of the algebra action and determines the composition series of these Specht modules, a novel analysis for hook bipartitions in this setting.

Findings

Explicit algebra generator action on basis elements.

Construction of irreducible submodules.

Complete determination of composition series.

Abstract

Brundan, Kleshchev and Wang equip the Specht modules $S_{\lambda}$ over the cyclotomic Khovanov--Lauda--Rouquier algebra $\mathscr{H}_n^{\Lambda}$ with a homogeneous $\mathbb{Z}$-graded basis. In this paper we begin the study of graded Specht modules labelled by hook bipartitions $((n-m),(1^m))$ in level $2$ of $\mathscr{H}_n^{\Lambda}$, which are precisely the Hecke algebras of type B, with quantum characteristic at least three. We give an explicit description of the action of the Khovanov--Lauda--Rouquier algebra generators $\psi_1,\dots,\psi_{n-1}$ on the basis elements of $S_{((n-m),(1^m))}$. Introducing certain Specht module homomorphisms, we construct irreducible submodules of these Specht modules, and thereby completely determining the composition series of Specht modules labelled by hook bipartitions for $e\geqslant{3}$.