Rui Xiong
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.
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The Canonical Bijection Between Pipe Dreams and Bumpless Pipe Dreams
本文利用无碰撞管道梦上的约化相容序列，构造了约化管道梦与约化无碰撞管道梦之间保持 Monk 公式的直接典范双射。
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。
We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams.
Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream.
Quantum bumpless pipe dreams
针对量子双舒伯特多项式缺乏组合表述的问题，提出量子无碰撞管道梦想的推广概念，给出其作为量子无碰撞管道梦想二项式权重之和的组合公式，并通过双射证明验证了公式正确性。
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。
In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018) introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of Lascoux (2002). Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguis…
We introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. As an application, we adopt Lenart's growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case.
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。
Recent work of Pechenik, Speyer, and Weigandt proved a formula for the degree of any Grothendieck polynomial. A distinct formula for the degree of vexillary Grothendieck polynomials was proven by Rajchgot, Robichaux, and Weigandt. We give a new proof of Pechenik, Speyer, and Weigandt's formula in the special case of vexillary permutations and characterize the set of bumpless pipe dreams which contribute maximal degree terms to the Grothendieck polynomial in this case. Furthermore, we use this characterization to draw connections between the Pechenik-Speyer-Weigandt and Rajchgot-Robichaux-Weiga…
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。
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。
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。
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。
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。
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…
We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Schützenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a "co-transition formula" which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the K-theoretic extensions.
Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical types.
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize …
Pipe Dream Rectification and Dual RSK Correspondence
该研究证明了 Dennin 关于对偶 RSK 对应变体对称性的猜想，通过超 Pipe Dream 修正与矩阵后缀向下归纳完成证明，明确了二元矩阵转置补的插入表与原矩阵记录表补的等式关系。
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。
Parametrizing the Grassmannian using pipe dreams
本文通过管道梦想参数化Deodhar组件，揭示了格拉斯曼流形的结构特性及求和项的额外信息。
Poset polytopes and pipe dreams: toric degenerations and beyond
本文利用管道梦想构造Dynkin类型A和C的标记链序多面体的环面退化，并由此引出PBW-单项式基与Newton--Okounkov体等对象。
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof.
The objects of interest in this thesis are positroid varieties in the Grassmannian, which are indexed by juggling patterns. In particular, we study affine patches on these positroid varieties. Our main result corresponds these affine patches to Kazhdan-Lusztig varieties in the affine Grassmannian. We develop a new term order and study how these spaces are related to subword complexes and Stanley-Reisner ideals. We define an extension of pipe dreams to the affine case and conclude by showing how our affine pipe dreams are generalizations of Cauchon and Le diagrams.
Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Sta…
Hybrid Pipe Dreams for Key Polynomials
本文为关键多项式构造了与Schubert多项式混合管道梦模型类似的新组合模型族。
Lattices of acyclic pipe dreams
本文证明出口排列为 $\omega$ 的无环管道梦递增翻转图是弱序区间 $[e,\omega]$ 的格商，并探讨其向有限 Coxeter 群子字复形无环面的推广。
Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams
本文研究双重Grothendieck多项式的Cauchy恒等式，提出pipe dream矫正算法构建pipe dreams间的权重保持双射，并基于新流算子揭示其对称性，最终恢复对偶RSK对应变体。
Lattices properties of acyclic alternating pipe dreams
本文针对A型Coxeter群子词复形的无环面，利用管道梦想表示证明了相关猜想，并给出所有无环面均为强无环的充分条件。
In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the h-polynomial of the corresponding pipe dream complex, which in certain cases equals the h-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to…
Derivatives of padded Schubert polynomials through pipe dreams
本文利用管道梦想的组合方法，为 Gaetz 和 Tung 关于对偶微分算子 $Δ$ 作用于 $π$-填充舒伯特多项式时的正性结果提供了一个新证明。
Poset polytopes and pipe dreams: types C and B
该研究利用偏序集多胞体与管道梦的组合联系，在C型和B型中构造并统一了表示基、Newton–Okounkov体及旗簇环面退化。
Flag positroid pipe dreams
引入旗正电子管图（FPPs），用于研究完全旗正电子，类似于Le-图在正电子研究中的作用，并展示其与Bruhat序区间、Richardson胞腔维数等组合性质的双射关系。
Crystal Chute Moves on Pipe Dreams
本文通过对 rc-图上的 chute moves 施加限制，在 Schubert 多项式的单项式上定义了 Demazure crystal 结构，并提供了将其分解为 key 多项式之和的方法。
A Proof of Rubey's Lattice Conjecture
本文证明了Rubey于2011年提出的关于约化管道梦偏序集为格的猜想，通过引入复合梯子移动操作$\mathcal{M}_{ij}$并推导递归公式，证明了并和交的存在性，同时给出了可比性的判定标准。
Hopf dreams and diagonal harmonics
本文在简化管道梦境上构造自由余自由的 Hopf 代数，并建立到排列 Hopf 代数的满射，其子代数关联 Catalan 数、Loday-Ronco 代数、格路及 ν-Tamari 格，并引入 Tamari 格中的 Hopf 链以应用于多元对角调和。
Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability
该研究基于旗簇K理论中β=1的格罗滕迪克多项式，通过映射到TASEP描述了约化管道梦想随机置换的极限置换子及n^(1/3)阶Tracy-Widom GUE涨落，还解决了相关开放问题及β=1格罗滕迪克多项式主特殊化最大值问题。
Random Subwords and Pipe Dreams
本文研究由相邻对换词经随机字母删除得到的随机排列，给出一大类起始词逆序数期望的渐近公式，并在特定词情形证明常数与 $n^{3/2}$ 增长，解决相关猜想。
Singular Points on Positroid Varieties and Planar N=4 Supersymmetric Yang-Mills Theory
该论文研究正合流形的奇点判定方法，给出两种计算T-不动点重数的途径，关联其与平面N=4超对称杨-米尔斯理论振幅的相关性，是2024年8月论文的修订版。
Random Permutations from Bott-Samelson Varieties
受随机管道梦模型启发，研究有限域上Bott-Samelson簇诱导的S_n上的概率分布，证明约化词R_1,R_2的分布相等当且仅当它们属于同一交换类。
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