Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913171102498816 |
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| author | Wahiche, David |
| author_facet | Wahiche, David |
| contents | We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_08071 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas Wahiche, David Combinatorics Mathematical Physics Representation Theory 05E05, 05E10, 05A15, 05A17, 11P81 We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald. |
| title | Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas |
| topic | Combinatorics Mathematical Physics Representation Theory 05E05, 05E10, 05A15, 05A17, 11P81 |
| url | https://arxiv.org/abs/2306.08071 |