Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908478830804992 |
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| author | Cain, Alan J. Malheiro, António Rodrigues, Fátima Rodrigues, Inês |
| author_facet | Cain, Alan J. Malheiro, António Rodrigues, Fátima Rodrigues, Inês |
| contents | Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a combinatorial study of these objects. We explain a previously-observed isomorphism of components of the quasi-crystal graph, and provide an explicit description using a new combinatorial structure called a quasi-array. Then two conjectures of Maas-Gariépy on the interaction of fundamental quasi-symmetric functions and Schur functions and on the arrangement of quasi-crystal components within crystal components are answered, the former positively, the latter negatively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions Cain, Alan J. Malheiro, António Rodrigues, Fátima Rodrigues, Inês Combinatorics 05E05 (Primary) 05E10 (Secondary) Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a combinatorial study of these objects. We explain a previously-observed isomorphism of components of the quasi-crystal graph, and provide an explicit description using a new combinatorial structure called a quasi-array. Then two conjectures of Maas-Gariépy on the interaction of fundamental quasi-symmetric functions and Schur functions and on the arrangement of quasi-crystal components within crystal components are answered, the former positively, the latter negatively. |
| title | Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions |
| topic | Combinatorics 05E05 (Primary) 05E10 (Secondary) |
| url | https://arxiv.org/abs/2309.14887 |