Schur-Sergeev duality for Ariki-Koike algebras

TL;DR

This paper establishes a duality between a quantized superalgebra and a cyclotomic Hecke algebra, extending Schur--Sergeev duality to a superalgebra setting with tensor representations.

Contribution

It introduces a new Schur--Weyl reciprocity for the superalgebra $U_q(rak{g})$ and cyclotomic Hecke algebra, generalizing classical dualities to superalgebra contexts.

Findings

Defined an action of $H_{m,n}(q,\mathbf{Q})$ on tensor space

Proved Schur--Weyl reciprocity between $U_q(\mathfrak{g})$ and $H_{m,n}(q,\mathbf{Q})$

Extended duality to superalgebra and cyclotomic Hecke algebra setting

Abstract

Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.