Double orthodontia formulas and Lascoux positivity
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929535727960064 |
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| author | Setiabrata, Linus Dizier, Avery St. |
| author_facet | Setiabrata, Linus Dizier, Avery St. |
| contents | We give a new formula for double Grothendieck polynomials based on Magyar's orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak S_w(\mathbf x;\mathbf y)$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1,\dots,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Double orthodontia formulas and Lascoux positivity Setiabrata, Linus Dizier, Avery St. Combinatorics We give a new formula for double Grothendieck polynomials based on Magyar's orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak S_w(\mathbf x;\mathbf y)$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1,\dots,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials. |
| title | Double orthodontia formulas and Lascoux positivity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.08038 |