论文标题
矩阵舒伯特品种的对角线变性
Diagonal degenerations of matrix Schubert varieties
论文作者
论文摘要
Knutson and Miller（2005）建立了基质Schubert品种的抗DiagonalGröbner变性与Pipe Dreams的现有组合之间的联系。他们使用这种对应关系为组合定义的舒伯特多项式作为Schubert类的代表提供了几何自然的解释。最近，Hamaker，Pechenik和Weigandt（2022）提出了与Matrix Schubert品种的对角线变性与浮力的Pipe Dreams，Lam，Lee和Shimozono（2021）引入的新型组合对象之间的类似联系。 Hamaker，Pechenik和Weigandt描述了基质Schubert品种定义理想的新生成集，并猜想了这些生成集形成对角线Gröbner碱基的排列表征。他们证明了这种猜想的特殊案例，并描述了在这些情况下，根据蓬松的烟斗梦想，矩阵舒伯特品种的对角线变性。本文的目的是证明完全普遍的猜想。该证明使用了与Rajchgot（2021）早期合作中建立的联络与几何顶点分解之间的联系。
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).
