Sergey Fomin, Andrei Zelevinsky
This paper focuses on the properties of Schubert cells as quasi-projective subvarieties of a generalized flag variety. More specifically, we investigate the problem of distinguishing between different Schubert cells using vanishing patterns of generalized Plucker coordinates.
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We briefly describe each of the four topics: Schubert Calculus, Schubert Cell, Schubert Cycle, and Schubert Polynomials.
A Pieri type formula for motivic Chern classes of Schubert cells in Grassmannians
本文证明了格拉斯曼流形中Schubert胞腔的motivic Chern类的Pieri公式，通过仿射Hecke代数转化计算，并应用于建立motivic Chern类与Segre motivic类的关系及寻找对偶层类的对称幂级数表示。
We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmative answer to a positivity conjecture of Aluffi and Mihalcea.
We discuss a relationship between Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds, Fomin-Kirillov algebra, and the generalized nil-Hecke algebra. We show that nonnegativity conjecture in Fomin-Kirillov algebra implies the nonnegativity of the Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds for type A. Motivated by this connection, we also prove that the (equivariant) Chern-Schwartz-MacPherson classes for Schubert cells in flag manifolds are certain summations of the structure constants of the equivariant cohomology of the Bott-Samelson varieties. W…
We prove that the number of connected components in the intersection of two open opposite Schubert cells in the variety of complete real n-dimensional flags equals 3*2^{n-1} for n>5.
We develop Pieri type as well as Murnaghan--Nakayama type formulas for equivariant Chern--Schwartz--MacPherson classes of Schubert cells in the classical flag variety. These formulas include as special cases many previously known multiplication formulas for Chern--Schwartz--MacPherson classes or Schubert classes. We apply the equivariant Murnaghan--Nakayama formula to the enumeration of rim hook tableaux.
We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of $d$-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of $d$.
A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.
We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of $\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we are able to systematically produce many negative answers to a question from the 1…
This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattic…
From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
本文证明了旗流形中Schubert胞的motivic Chern类的多个性质与猜想，利用motivic Chern变换定义了等变Hirzebruch变换，并借助Demazure-Lusztig递归给出了Hirzebruch类及相关对偶的递归公式。
The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits …
Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical Grassmannian manifold. It has a Schubert cell decomposition indexed by the coroot lattice or equivalently by the minimal length coset representatives for the affine Weyl group modulo the Weyl group for G. The closure of an affine Schubert cell is a finite dimensional projective variety that we call an affine Schubert variety. In this paper we completely dete…
Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
本文比较了偏旗簇中开放投影Richardson簇与仿射Schubert胞腔的Segre--MacPherson类，并在Grassmannian情形下，利用仿射排列的pipe dreams显式构造了表示开放positroid簇该类的对称函数。
Let $\mathbb{F}$ be any field. The Grassmannian $\mathrm{Gr}(m,n)$ is the set of $m$-dimensional subspaces in $\mathbb{F}^n$, and the general linear group $\mathrm{GL}_n(\mathbb{F})$ acts transitively on it. The Schubert cells of $\mathrm{Gr}(m,n)$ are the orbits of the Borel subgroup $\mathcal{B} \subset \mathrm{GL}_n(\mathbb{F})$ on $\mathrm{Gr}(m,n)$. We consider the association scheme on each Schubert cell defined by the $\mathcal{B}$-action and show it is symmetric and it is the generalized wreath product of one-class association schemes, which was introduced by R. A. Bailey [European Jou…
Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes
本文基于Lenart和Postnikov的λ链，推导了广义旗 manifold 中 Schubert 胞 motivic Chern 类与线丛类相乘的 Chevalley 公式，并探讨了稳定包在 T*(G/B) 中的墙穿越公式及多项式系数的正性性质。
Motivic Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells
本文研究了开放投影里奇变种和仿射Schubert单元的motivic Chern类，通过递推关系和格拉斯曼流形中的正体种公式，建立了与扭曲Kazhdan–Lusztig R多项式的关系。
In this article, the comodule structure of Chow rings of Flag manifolds $\operatorname{CH}(G/B)$ is described by Schubert cells. Its equivariant version gives rise to a Hopf structure of the equivariant cohomology of flag manifolds $H^*_B(G/B)$. We get two identities of generalized Schubert polynomials as explanations of the geometric facts.
A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have identical Schubert cell structures. This also produces a complete classification of isomorphism types of lower intervals of 312-avoiding permutations in the Bruhat order.
Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable.
Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem
本文计算了旗流形中 Schubert 胞的 motivic Chern 类，证明其由 Hecke 代数中的 Demazure-Lusztig 算子递归决定，并利用此递归与 Langlands 对偶性证明了关于标准基与 Casselman 基过渡矩阵系数分解及全纯性质的两个猜想。
Partial order on involutive permutations and double Schubert cells
本研究发现对称群中对合的偏序（源于Borel子群作用于平方零上三角矩阵的轨道闭包含序），与两个格拉斯曼流形直积中B-轨道闭包含序对应的对合偏序一致，并建立了二者的几何联系。
Motivic Segre classes of Schubert cells and the connective formal group law
利用连接形式群律定义 d 步旗簇中 Schubert 胞腔的动机 Segre 类的单参数 β 形变，并给出 d=1 时的有理函数表示和组合公式。
Schubert cells of mixed type in complex Lagrangian Grassmannians
本文描述了复 Lagrange Grassmann 流形的 CW 分解，显式计算了粘贴映射的度，并由此得出了实与复 Lagrange Grassmann 流形的同伦扩张性质及部分挠类的可缩性。
Cell closures for two-row Springer fibers via noncrossing matchings
本文研究两行 Springer 纤维的胞腔闭包，通过定义“切割弧”操作显式描述 Springer Schubert 胞腔的闭包关系，证明其边界由所有切割弧方式对应的仿射子集构成。
Using combinatorial properties of symmetric polynomials, we compute explicitly the Soergel modules for some permutations whose corresponding Schubert varieties are rationally smooth. We build from them diagram algebras whose module categories are equivalent to the subquotient categories of the BGG category $\mathcal{O}(\mathfrak{gl}_n)$ which show up in categorification of $\mathfrak{gl}(1|1)$-representations. We construct diagrammatically the graded cellular structure and the properly stratified structure of these algebras.
Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
本文将可构造函数的等变CSM类刻画为余切丛零截面下的特征闭链限制，利用Verma模与稳定包络的等价性证明了旗流形中Schubert胞CSM类的Schubert基展开具有正性，验证了Aluffi-Mihalcea猜想。
We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent stu…
Quasisymmetric Schubert calculus
本文在Baker和Richter工作基础上，将舒伯特演算思想应用于圈空间的James约化积，构造其上同调的拟对称舒伯特基，还研究了更精细的K-理论舒伯特基，解决了拟对称函数理论缺乏舒伯特演算类比的问题。
Grove polynomials and $K$-theoretic quasisymmetry
本文定义了grove多项式，作为森林多项式的集合值扩展，并展示了它们在$K$-理论下与准对称Schubert单元 dual，类似于Grothendieck多项式与完全旗形簇中Schubert单元的关系。作为结果，Lam-Pylyavskyy的多基本准对称函数的有限截断获得了几何解释，作为准对称Schubert单元的$K$-理论代表。
A Gröbner basis for Kazhdan-Lusztig ideals of the flag variety of affine type A
本文通过Bott-Samelson映射对仿射A型旗流形中的Schubert胞进行线性参数化，给出了生成Kazhdan-Lusztig理想的显式方程，并证明其构成Gröbner基，推广了Woo-Yong在A型旗流形中的结果。
Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell
本文基于 Peterson 未发表结果，通过推广 Ciocan-Fontanine 和 Givental-Kim 的量子上同调环表示，给出正则幂零 Hessenberg 簇与开相反舒伯特胞交的坐标环刻画，并应用于奇异轨迹分析及循环商奇点关联研究。
It is well known that many geometric properties of Schubert varieties of type $A$ can be interpreted combinatorially. Given two permutations $w,x\in S_n$ we give a combinatorial consequence of the property that the smooth locus of the Schubert variety $X_w$ contains the Schubert cell $Y_x$. This is a necessary ingredient for the interpretation of recent representation-theoretic results of the author with Mínguez in terms of identities of Kazhdan--Lusztig polynomials.
James reduced product schemes and double quasisymmetric functions
该研究围绕James约化积$J\mathbb{C}\mathbb{P}^\infty$展开，刻画其环面等变上同调的胞腔基与对应双重单项式拟对称函数的组合规则，还证明复射影簇的James约化积仍为射影簇，推动拟对称函数的几何理论发展。
Parametrization of positroid cells by Dubrovin-Natanzon Cartier divisors on maximal Mumford curves
该研究证明全正Schubert胞是对应Le图的极大Mumford曲线雅可比簇的实分支，并构造了正合胞点与该曲线上Dubrovin-Natanzon除子间的双射。
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