On the relationship between dominance order and $ θ $-dominance order on multipartitions - Kai Zhou (AIMS Mathematics)
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 $ \{ψ_{{\mathfrak{s}}{\mathfrak{t}}}\} $. With $ θ $-dominance order on multipartitions, Bowman constructed integral cellular basis $ \{c^{θ}_{{\mathfrak{s}}{\mathfrak{t}}}\} $. 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 $ θ $-dominance order on multipartitions. In this paper, we define the weak $ θ $-dominance order and give a combinatorial description of the neighbors with weak $ θ $-dominance order. Then we prove weak $ θ $-dominance order is equivalent to dominance order whenever the loading $ θ $ is strongly separated. As a corollary, we give the relationship between weak $ θ $-dominance order, $ θ $-dominance order, and dominance order on multipartitions.