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^{\theta}_{{\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 $ \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.
Citation: Kai Zhou. On the relationship between dominance order and $ \theta $-dominance order on multipartitions[J]. AIMS Mathematics, 2025, 10(2): 2998-3012. doi: 10.3934/math.2025139
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^{\theta}_{{\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 $ \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.
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Kai Zhou. On the relationship between dominance order and $ \theta $-dominance order on multipartitions[J]. AIMS Mathematics, 2025, 10(2): 2998-3012. doi: 10.3934/math.2025139
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