@article{Zhou2025, 
author = {Kai Zhou},
title = {On the relationship between dominance order and    θ-dominance order on multipartitions},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {2},
pages = {2998-3012},
keywords = {cyclotomic Hecke algebras, multipartitions, dominance order, θ-dominance order, weak θ-dominance order},
url = {https://www.sciopen.com/article/10.3934/math.2025139},
doi = {10.3934/math.2025139},
abstract = {Many cellular bases have been constructed for the cyclotomic Hecke algebras of    G  (  ℓ  ,  1  ,  n  ). For example, with dominance order on multipartitions, Dipper, James, and Mathas constructed a cellular basis    {      m                            s                                      t                      } and Hu, Mathas constructed a graded cellular basis    {      ψ                            s                                      t                      }. With    θ-dominance order on multipartitions, Bowman constructed integral cellular basis    {      c                            s                                      t                            θ        }. 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  (  ℓ  ,  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.}
}