Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914623431639040 |
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| author | Moura, Guilherme Zeus Dantas e Mandelshtam, Olya |
| author_facet | Moura, Guilherme Zeus Dantas e Mandelshtam, Olya |
| contents | Permuted-basement Macdonald polynomials $E_α^σ(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $σ$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E_α^σ$, $E_α^{σs_i}$, $E_{s_iα}^σ$, and $E_{s_iα}^{σs_i}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E_α^σ$ in the polynomials $\{E_{s_iα}^τ\}_τ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02395 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials Moura, Guilherme Zeus Dantas e Mandelshtam, Olya Combinatorics 05E05, 05A19, 33D52 Permuted-basement Macdonald polynomials $E_α^σ(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $σ$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E_α^σ$, $E_α^{σs_i}$, $E_{s_iα}^σ$, and $E_{s_iα}^{σs_i}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E_α^σ$ in the polynomials $\{E_{s_iα}^τ\}_τ$. |
| title | Shape changing identities for permuted-basement nonsymmetric Macdonald polynomials |
| topic | Combinatorics 05E05, 05A19, 33D52 |
| url | https://arxiv.org/abs/2606.02395 |