矩阵Schubert簇的对角退化
Diagonal degenerations of matrix Schubert varieties
摘要：
Knutson和Miller（2005）建立了矩阵舒伯特簇的反对角Gröbner退化与先前存在的白日梦组合学之间的联系。他们利用这种对应关系给出了一个几何上自然的解释，解释了组合定义的舒伯特多项式作为舒伯特类的代表。最近，Hamaker，Pechenik和Weigandt（2022）提出了矩阵舒伯特簇的对角退化与无扰白日梦之间的类似联系，Lam，Lee和Shimozono（2021）介绍了较新的组合对象。Hamaker、Pechenik和Weigandt描述了矩阵Schubert簇的定义理想的新生成集，并描述了这些生成集形成对角Gröbner基的置换的特征。他们证明了这一猜想的特殊情况，并在这些情况下描述了矩阵舒伯特品种的对角退化。本文的目的是充分一般地证明这个猜想。该证明使用了Rajchgot（2021）早期工作中建立的联络和几何顶点分解之间的联系。
英文摘要：
Knutson and Miller (2005) established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt (2022) proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono (2021). Hamaker, Pechenik, and Weigandt described new generating sets of the defining ideals of matrix Schubert varieties and conjectured a characterization of permutations for which these generating sets form diagonal Gröbner bases. They proved special cases of this conjecture and described diagonal degenerations of matrix Schubert varieties in terms of bumpless pipe dreams in these cases. The purpose of this paper is to prove the conjecture in full generality. The proof uses a connection between liaison and geometric vertex decomposition established in earlier work with Rajchgot (2021).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
