单、双与三Schubert结构常数的非零性
Non-vanishing of Single, Double, and Triple Schubert Structure Constants
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中文总结 AI 辅助
本文研究双、三Schubert系数的非零性问题，证明三Schubert系数非零性由单Schubert系数决定，得到三Littlewood--Richardson系数的饱和性质，提出双Schubert系数非零性相关猜想并证明部分情形成立。
中文摘要 AI 辅助
Schubert零化问题旨在探究单Schubert系数$c_{u,v}^w$是否为零。本文研究了双Schubert系数$c_{u,v}^w(t)$与三Schubert系数$c_{u,v}^w(t;y)$的非零性问题。我们证明，$c_{u,v}^w(t;y)$的非零性完全由单Schubert系数的非零性决定。作为副产品，我们得到了三Littlewood--Richardson系数$c_{λ,μ}^ν(t;y)$的饱和性质。此外，我们提出一个猜想，认为$c_{u,v}^w(t)$的非零性也由单或三Schubert系数的非零性决定。我们证明了该猜想的一个方向包含关系；对于反向包含，我们证明猜想在以下三种情形下成立：Pieri情形、分离下降情形与逆Grassmannian情形。
英文摘要
The Schubert vanishing problem asks whether the single Schubert coefficients $c_{u,v}^w$ are zero. In this paper, we consider the non-vanishing problems of double Schubert coefficients $c_{u,v}^w(t)$ and triple Schubert coefficients $c_{u,v}^w(t;y)$. We show that the non-vanishing of $c_{u,v}^w(t;y)$ is completely determined by the non-vanishing of single Schubert coefficients. As a byproduct, we obtain the saturation property of the triple Littlewood--Richardson coefficients $c_{λ,μ}^ν(t;y)$. Moreover, we pose a conjecture asserting that the non-vanishing of $c_{u,v}^w(t)$ is also determined by the non-vanishing of single or triple Schubert coefficients. We prove a one-side inclusion of the conjecture. For the reverse inclusion, we show that the conjecture holds for the following three cases: the Pieri case, the separated descents case, and the inverse Grassmannian case.
