A Molev-Sagan type formula for double Schubert polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909119681658880 |
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| author | Samuel, Matthew J. |
| author_facet | Samuel, Matthew J. |
| contents | We give a Molev-Sagan type formula for computing the product $\mathfrak{S}_u(x;y)\mathfrak{S}_v(x;z)$ of two double Schubert polynomials in different sets of coefficient variables where the descents of $u$ and $v$ satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial $\mathfrak{S}_u(x;y)$ by a factorial elementary symmetric polynomial $E_{p,k}(x;z)$. Both formulas remain positive in terms of the negative roots when we set $y=z$, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial $s_λ(x_1,\ldots,x_m;y)$ by a double Schubert polynomial $\mathfrak{S}_v(x_1,\ldots,x_p;y)$ such that $m\geq p$. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Molev-Sagan type formula for double Schubert polynomials Samuel, Matthew J. Combinatorics Algebraic Geometry 05E05, 05E14, 05A15, 05A05 (Primary) 14N10 (Secondary) We give a Molev-Sagan type formula for computing the product $\mathfrak{S}_u(x;y)\mathfrak{S}_v(x;z)$ of two double Schubert polynomials in different sets of coefficient variables where the descents of $u$ and $v$ satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial $\mathfrak{S}_u(x;y)$ by a factorial elementary symmetric polynomial $E_{p,k}(x;z)$. Both formulas remain positive in terms of the negative roots when we set $y=z$, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial $s_λ(x_1,\ldots,x_m;y)$ by a double Schubert polynomial $\mathfrak{S}_v(x_1,\ldots,x_p;y)$ such that $m\geq p$. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials. |
| title | A Molev-Sagan type formula for double Schubert polynomials |
| topic | Combinatorics Algebraic Geometry 05E05, 05E14, 05A15, 05A05 (Primary) 14N10 (Secondary) |
| url | https://arxiv.org/abs/2401.11060 |