Vexillary Grothendieck Polynomials via Bumpless Pipe Dreams

Puzzle Model for Bumpless Pipe Dream

In this note, a new puzzle is introduced where the pipe dream and bumpless pipe dream can be played simultaneously. Using these, a combinatorial proof of the (ordinary) Schubert polynomials in terms of bumpless pipe dream is given. The main tool is the Yang--Baxter equation.

弱序：交错符号矩阵、单调三角形和无碰撞管道梦想

Weak order: Alternating sign matrices, monotone triangles, and bumpless pipe dreams

本文证明了交错符号矩阵上弱序的两种定义等价，并给出了三种计算覆盖关系的方法，最后用无碰撞管道梦想刻画了弱序算子的纤维。

无凸管梦碎片 -- ($GL_p\times GL_q$)-轨道的等变几何

Bumpless Pipe Dream Fragments -- Equivariant Geometry of Clans

本文通过定义clan多项式，解决了($GL_p\times GL_q$)-轨道闭包基本类等变Schubert展开系数的问题。

关于完全对称自补平面划分的管道梦视角

A pipe dream perspective on totally symmetric self-complementary plane partitions

本文通过管道梦视角刻画完全对称自补平面划分，建立其与交替符号矩阵的双射，并保持偏序结构。

无碰撞管道梦想与Puzzles的相遇

Bumpless pipe dreams meet Puzzles

本文提出了一种基于pipe puzzles的公式，用于展开双Grothendieck多项式乘积的三重Schubert微积分版本，并借此恢复了已有公式与模型，部分证实了Kirillov的正性猜想。

无碰撞管道梦编码 Schubert 多项式的 Gröbner 几何

Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials

本文证明无碰撞管道梦可按对角项序为矩阵 Schubert 簇的对角 Gröbner 退化不可约分量索引，证实相关猜想并确立其与经典管道梦的对偶几何地位。

Double Grothendieck polynomials and colored lattice models

We construct an integrable colored six-vertex model whose partition function is a double Grothendieck polynomial. This gives an integrable systems interpretation of bumpless pipe dreams and recent results of Weigandt [arXiv:2003.07342] relating double Grothendieck polynomias with bumpless pipe dreams. For vexillary permutations, we then construct a new model that we call the semidual version model. We use our semidual model and the five-vertex model of Motegi and Sakai to given a new proof that double Grothendieck polynomials for vexillary permutations are equal to flagged factorial Grothendie…

Schubert RSK 的增长图

Growth Diagrams for Schubert RSK

本文为 Huang–Pylyavskyy 在 plactic 双字上建立的广义 RSK 对应构造了 Fomin 增长图的类比，并证明其可恢复 Gao–Huang 的管道梦与无碰管道梦之间的双射。

用 Pipe Dream 组合学进行基变换

Changing Bases with Pipe Dream Combinatorics

本文利用无碰撞管道梦 BPD 建立 Schubert 与 Grothendieck 多项式互展的显式组合规则，并推广到后稳定情形。

Gröbner geometry of Schubert polynomials through ice

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of "vexillary'' matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by Lam, Lee, and S…

通用管道梦想、上下变体与Schwartz-MacPherson类

Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

本文回顾了上下变体，给出了基于通用管道梦想求和的等变上同调类公式，恢复了双重Schubert多项式的经典与无缝管道梦想公式，并计算了交换簇度数及特定轨道的Chern-Schwarz-MacPherson类。

上下方案的混合管道梦

Hybrid pipe dreams for the lower-upper scheme

本文推广混合管道梦至通用情形，用基于杨-巴克斯特的方法证明等变下的混合独立性与递推关系，其多项式可计算上下簇的等变类，并给出两种证明，还提出通量定义以推导管道梦概念。

Schubert特化的计算与采样

Computation and sampling for Schubert specializations

本文研究Schubert多项式主特化计数减少管道幻想和减少无管幻想，发现Merzon-Smirnov猜想在n=17时的第一个反例，并探讨了与Grothendieck多项式相似的渐进行为。