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| Format: | Preprint |
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2022
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| Accès en ligne: | https://arxiv.org/abs/2206.05177 |
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| _version_ | 1866909451778260992 |
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| author | Lapointe, Luc |
| author_facet | Lapointe, Luc |
| contents | We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric Schur functions through a somewhat complicated process involving their dual basis, tableaux combinatorics, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the $m$-symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of $m$-symmetric Schur functions. We obtain relations on the $(q,t)$-Koska coefficients $K_{ΩΛ}(q,t)$ in the $m$-symmetric world, and show in particular that the usual $(q,t)$-Koska coefficients are special cases of the $K_{ΩΛ}(q,t)$'s. Finally, we show that when $m$ is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05177 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures Lapointe, Luc Combinatorics 05E05 We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric Schur functions through a somewhat complicated process involving their dual basis, tableaux combinatorics, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the $m$-symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of $m$-symmetric Schur functions. We obtain relations on the $(q,t)$-Koska coefficients $K_{ΩΛ}(q,t)$ in the $m$-symmetric world, and show in particular that the usual $(q,t)$-Koska coefficients are special cases of the $K_{ΩΛ}(q,t)$'s. Finally, we show that when $m$ is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials. |
| title | $m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2206.05177 |