Combinatorial proof of a permuted basement Macdonald polynomial identity
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866912557622624256 |
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| author | Orr, Daniel Rivera Jr, Johnny |
| author_facet | Orr, Daniel Rivera Jr, Johnny |
| contents | A well-known and fundamental property of the Macdonald polynomials $P_λ(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial proof of a permuted basement Macdonald polynomial identity Orr, Daniel Rivera Jr, Johnny Combinatorics Quantum Algebra Representation Theory A well-known and fundamental property of the Macdonald polynomials $P_λ(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution. |
| title | Combinatorial proof of a permuted basement Macdonald polynomial identity |
| topic | Combinatorics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2508.20337 |