On the relationship between dominance order and $\theta$-dominance order on multipartitions

Many cellular bases have been constructed for the cyclotomic Hecke algebras of $G(\ell ,1,n)$. For example, with dominance order on multipartitions, Dipper, James, and Mathas constructed a cellular basis $\{m_{\mathfrak{s}\mathfrak{t}}\}$ and Hu, Mathas constructed a graded cellular basis $\{{\psi }_{\mathfrak{s}\mathfrak{t}}\}$. With $\theta$-dominance order on multipartitions, Bowman constructed integral cellular basis $\{c_{\mathfrak{s}\mathfrak{t}}^{\theta }\}$. Following Graham and Lehrer's cellular theory, different constructions of cellular basis may determine different parameterizations of simple modules of the cyclotomic Hecke algebras of $G(\ell ,1,n)$. To study the relationship between these parameterizations, it is necessary to understand the relationship between dominance order and $\theta$-dominance order on multipartitions. In this paper, we define the weak $\theta$-dominance order and give a combinatorial description of the neighbors with weak $\theta$-dominance order. Then we prove weak $\theta$-dominance order is equivalent to dominance order whenever the loading $\theta$ is strongly separated. As a corollary, we give the relationship between weak $\theta$-dominance order, $\theta$-dominance order, and dominance order on multipartitions.