Specht modules labelled by hook bipartitions I

Abstract

Brundan, Kleshchev and Wang equip the Specht modules $S_{\lambda }$ over the cyclotomic Khovanov--Lauda--Rouquier algebra ${\mathcal{H}}_n^{\Lambda }$ 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 ${\mathcal{H}}_n^{\Lambda }$, 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 $e⩾3$.

v2: shorter version, clarified Proposition 6.15

References in corpus (5)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Graded decomposition numbers for cyclotomic Hecke algebras
• Generalised column removal for graded homomorphisms between Specht modules
• On bases of some simple modules of symmetric groups and Hecke algebras

Cited by in corpus (1)

• Decomposable Specht modules indexed by bihooks