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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.20622 |
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| _version_ | 1866909900048695296 |
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| author | Hao, Lingxiao Zhu, Shenglin |
| author_facet | Hao, Lingxiao Zhu, Shenglin |
| contents | John M. Campbell constructed a combinatorial Hopf algebra (CHA) \text{ParSym} on partition diagrams by lifting the CHA structure of \text{NSym} (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define \text{ParQSym}, which is the graded dual of \text{ParSym}. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA \text{QSym} of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of \text{ParQSym}, some gradings and filtrations of \text{ParSym} and \text{ParQSym}, and some bases of \text{ParSym} and \text{ParQSym} by analogy with some distinguished bases of \text{NSym} and \text{QSym}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20622 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams Hao, Lingxiao Zhu, Shenglin Rings and Algebras John M. Campbell constructed a combinatorial Hopf algebra (CHA) \text{ParSym} on partition diagrams by lifting the CHA structure of \text{NSym} (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define \text{ParQSym}, which is the graded dual of \text{ParSym}. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA \text{QSym} of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of \text{ParQSym}, some gradings and filtrations of \text{ParSym} and \text{ParQSym}, and some bases of \text{ParSym} and \text{ParQSym} by analogy with some distinguished bases of \text{NSym} and \text{QSym}. |
| title | The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2504.20622 |