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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2001.03922 |
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| _version_ | 1866917614774648832 |
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| author | Fan, Neil J. Y. Guo, Peter L. Liu, Nicolas Y. |
| author_facet | Fan, Neil J. Y. Guo, Peter L. Liu, Nicolas Y. |
| contents | Let $\mathfrak{S}_w(x)$ be the Schubert polynomial for a permutation $w$ of $\{1,2,\ldots,n\}$. For any given composition $μ$, we say that $x^μ\mathfrak{S}_w(x^{-1})$ is the complement of $\mathfrak{S}_w(x)$ with respect to $μ$. When each part of $μ$ is equal to $n-1$, Huh, Matherne, Mészáros and St.\,Dizier proved that the normalization of $x^μ\mathfrak{S}_w(x^{-1})$ is a Lorentzian polynomial. They further conjectured that the normalization of $\mathfrak{S}_w(x)$ is Lorentzian. It can be shown that if there exists a composition $μ$ such that $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then the normalization of $\mathfrak{S}_w(x)$ will be Lorentzian. This motivates us to investigate the problem of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial. We show that if $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then $μ$ must be a partition. We also consider the case when $μ$ is the staircase partition $δ_n=(n-1,\ldots, 1,0)$, and obtain that $x^{δ_n} \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial if and only if $w$ avoids the patterns 132 and 312. A conjectured characterization of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial is proposed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_03922 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Complements of Schubert polynomials Fan, Neil J. Y. Guo, Peter L. Liu, Nicolas Y. Combinatorics Let $\mathfrak{S}_w(x)$ be the Schubert polynomial for a permutation $w$ of $\{1,2,\ldots,n\}$. For any given composition $μ$, we say that $x^μ\mathfrak{S}_w(x^{-1})$ is the complement of $\mathfrak{S}_w(x)$ with respect to $μ$. When each part of $μ$ is equal to $n-1$, Huh, Matherne, Mészáros and St.\,Dizier proved that the normalization of $x^μ\mathfrak{S}_w(x^{-1})$ is a Lorentzian polynomial. They further conjectured that the normalization of $\mathfrak{S}_w(x)$ is Lorentzian. It can be shown that if there exists a composition $μ$ such that $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then the normalization of $\mathfrak{S}_w(x)$ will be Lorentzian. This motivates us to investigate the problem of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial. We show that if $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then $μ$ must be a partition. We also consider the case when $μ$ is the staircase partition $δ_n=(n-1,\ldots, 1,0)$, and obtain that $x^{δ_n} \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial if and only if $w$ avoids the patterns 132 and 312. A conjectured characterization of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial is proposed. |
| title | Complements of Schubert polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2001.03922 |