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 $...
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AIMS Press
2025-02-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025139 |
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| author | Kai Zhou |
| author_facet | Kai Zhou |
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| description | 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. |
| format | Article |
| id | doaj-art-c7c1f86119e84cedb15fbc362afdfbdb |
| institution | OA Journals |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-c7c1f86119e84cedb15fbc362afdfbdb2025-08-20T01:54:41ZengAIMS PressAIMS Mathematics2473-69882025-02-011022998301210.3934/math.2025139On the relationship between dominance order and $ \theta $-dominance order on multipartitionsKai Zhou0School of Science, Shandong Jianzhu University, Jinan, 250101, ChinaMany 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 $-dominance order, and dominance order on multipartitions.https://www.aimspress.com/article/doi/10.3934/math.2025139cyclotomic hecke algebrasmultipartitionsdominance order$ \theta $-dominance orderweak $ \theta $-dominance order |
| spellingShingle | Kai Zhou On the relationship between dominance order and $ \theta $-dominance order on multipartitions AIMS Mathematics cyclotomic hecke algebras multipartitions dominance order $ \theta $-dominance order weak $ \theta $-dominance order |
| title | On the relationship between dominance order and $ \theta $-dominance order on multipartitions |
| title_full | On the relationship between dominance order and $ \theta $-dominance order on multipartitions |
| title_fullStr | On the relationship between dominance order and $ \theta $-dominance order on multipartitions |
| title_full_unstemmed | On the relationship between dominance order and $ \theta $-dominance order on multipartitions |
| title_short | On the relationship between dominance order and $ \theta $-dominance order on multipartitions |
| title_sort | on the relationship between dominance order and theta dominance order on multipartitions |
| topic | cyclotomic hecke algebras multipartitions dominance order $ \theta $-dominance order weak $ \theta $-dominance order |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025139 |
| work_keys_str_mv | AT kaizhou ontherelationshipbetweendominanceorderandthetadominanceorderonmultipartitions |