Specht modules labelled by hook bipartitions I

Abstract

Brundan, Kleshchev and Wang equip the Specht modules $S_λ$ over the cyclotomic Khovanov--Lauda--Rouquier algebra $\mathscr{H}_n^Λ$ 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 $\mathscr{H}_n^Λ$, 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 $ψ_1,\dots,ψ_{n-1}$ 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 $e\geqslant{3}$.