Orthogonal divided differences and Schubert polynomials, P-functions, and vertex operators
Alain Lascoux, Piotr Pragacz
本文给出了几个新的公式，这些公式对于与正交群及相关的正交退化轨迹有关的Schubert演算是有用的。
We give several new formulas which are useful for Schubert Calculus associated with the orthogonal groups and related orthogonal degeneracy loci.
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Coefficient lifting for skew divided differences and triple Schubert coefficients
本文研究斜分差算子的系数提升，证明其作用于双舒伯特多项式的单项式系数对应三重舒伯特系数，应用该结论为Kirillov的非负猜想提供了新证明。
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.
We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(λ_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $ω$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted se…
Graham positivity of triple Schubert calculus
本文通过建立精细化 Graham 正性定理，证明 Samuel 关于三组变量中两个双 Schubert 多项式展开系数正性的猜想，并推出 Kirillov 关于斜除以差分算子作用于 Schubert 多项式正性的猜想。
Long range divided differences, clusters, and Graham-positivity
本文通过长程差商运算和组合算法，将环面轨道闭包的等变同调类展开为Graham正Schubert循环组合，推广了AJS-Billey公式并应用于曲线、轨道闭包及Richardson簇。
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.
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
We prove that twisted versions of Schubert polynomials defined by $\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and $\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the $\widetilde{\mathfrak S}_w$ as well as their localizations.
Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians
我们计算的初始理想，关于某些方便地选择的条款的订单，切锥的理想在环面不动点的舒伯特品种在正交格拉斯曼。初始理想是无平方的单项理想，因此是单纯复形的Stanley-Reisner面环。我们描述了这些复合物。这些复合物的最大面编码某些不相交的晶格路径集。
We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel definition of double and triple Stanley symmetric functions; 3) a proof of the positivity of double Edelman-Greene coefficients generalizing the results of Edelman-Greene and Lascoux-Schutzenberger; 4) the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form o…
We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian a…
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, f…
Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the singular and quantum cohomology ring of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calculus on even and odd orthogonal Grassmannians. We also study eta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X.
Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians
给出了以下问题的解决方案：如何计算在正交格拉斯曼中的舒伯特簇上的点处的重数，或更一般地计算希尔伯特函数。应用标准单项式理论将问题从几何转化为组合数学。由此产生的组合问题的解决方案形成了大部分的文件。这种方法已经遵循早先解决同样的问题的格拉斯曼和辛格拉斯曼。作为一个应用，我们提出了一个解释的多重性的数量不相交的格子路径的某种。以舒伯特簇是一种特殊的类型和点是“单位陪集”，我们的问题专门的问题Pfedian理想治疗的不同方法存在于文献中。在文献中也可得到当点是“一般奇点”时的几何解。"
Rigid Schubert classes in partial flag varieties
本文研究 A 型偏旗簇及 B、D 型正交偏旗簇中 Schubert 类的刚性，并给出判定其刚性的数值条件。
The Rigidity Problem in Orthogonal Grassmannians
本文分类了正交Grassmann流形中的刚性Schubert类，并给出了保证由Schubert类代表元参数化的线性空间与固定线性空间在所需维度相交的组合条件。
General isotropic flags are general (for Grassmannian Schubert calculus)
我们表明，一般各向同性旗奇正交和辛群是一般的Schubert演算的经典格拉斯曼在Schubert细胞定义的这样的标志满足横向。这加强了Belkale和Kumar在arXiv：0708.0398中的结果。
A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety $X^\gc$ in the Orthogonal Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any $T$-fixed point of $X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal bounded RSK (OBRSK). The…
We describe a relationship between work of Laksov, Gatto, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirković and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the "rim-hook" rule for computing in the equivariant quantum cohomology of the Grassmannian.
We prove an analogue of the Mukhin-Tarasov-Varchenko theorem (formerly the Shapiro-Shapiro conjecture) for the maximal type B_n orthogonal Grassmannian OG(n,2n+1).
We obtain an algorithm computing the Chern-Schwartz-MacPherson (CSM) classes of Schubert cells in a generalized flag manifold G/B. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure-Lusztig type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of G/B. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold G/P. We conjecture that the CSM classes of Sch…
We examine the relationship between the (double) Schubert polynomials of Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl group invariants. We obtain generators for the kernel of the natural map from the corresponding ring of Schubert polynomials to the (equivariant) cohomology ring of symplectic and orthogonal flag manifolds.
Double Schubert polynomials and degeneracy loci for the classical groups
本文提出了B，C，D型李的双Schubert多项式P_w（X，Y）理论，它自然地推广了A型Schutzenberger的Lascoux多项式族。这些多项式满足正性、正交性和稳定性，并代表了向量丛的Schubert簇和退化轨迹类。当w是外尔群的极大Grassmannian元时，P_w（X，Y）可以用Schur型行列式和Pfirmans表示，类似于Kempf和Laksov的A型公式.由量子上同调激发的一个例子表明，丛的“各向同性态射”的退化轨迹不存在陈类公式。
In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case. T…
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as i…
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand-Zetlin polytopes and twisted cubes of Grossberg-Karshon are obtained in a uniform way.
We present a proof of a Littlewood-Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured in [Thomas-Yong '09]. Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, [Buch-Ravikumar '09] proved a Pieri rule for OG(n,2n+1) that [Feigenbaum-Sergel '09] showed confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.
Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory
该研究探讨部分旗流形上左乘诱导的Demazure-Lusztig算子，证明其生成Schubert胞腔的相关特征类，并构造出满足量子积相容Leibniz法则的等变量子左除差算子。
Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{α_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n}.$ Let $X$ be a Schubert variety in $G/P^{α_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2ω_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{α_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence,…
We consider the loci of invertible linear maps $f : \mathbb{C}^n \to {(\mathbb{C}^n)}^*$ together with pairs of flags $(E_\bullet, F_\bullet)$ in $\mathbb{C}^n$ such that the various restrictions $f : F_j \to E_i^*$ have specified ranks. Identifying an invertible linear map with its graph viewed as a point in a Grassmannian, we show that the closures of these loci have cohomology classes represented by the back-stable Schubert polynomials of Lam, Lee, and Shimozono. As a special case, we recover the result of Knutson, Lam, and Speyer that Stanley symmetric functions represent the classes of gr…
In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter ($\hbar$) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov's stable envelope notion. We review formulas for the $\hbar$-deformed classes originating from Tarasov-Varchenko weight functions, as well as their orthogonality relations. As a consequence, explici…
The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
本文计算置换体簇上同调类在舒伯特类基下的展开，证明结构常数为正，并给出其组合解释及推广到其他李类型的思路。
We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. We show that the cohomology of the affine flag variety is product of the cohomology of an affine Grassmannian and a flag variety, which are generated by MN elements and Dunkl…
We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for …
We study the double Grothendieck polynomials of Kirillov--Naruse for the symplectic and odd orthogonal Grassmannians. These functions are explicitly written as sums of Pfaffian and are identified with the stable limits of the fundamental classes of Schubert varieties in the torus equivariant connective K-theory of these isotropic Grassmannians. We also provide a combinatorial description of the ring formally spanned by double Grothendieck polynomials.
Algebra and geometry of ASM weak order
该研究将ASM上的弱布鲁哈特序与代数几何结合，刻画了ASM簇的余维数，证明弱序算子与K理论差分算子可交换且满足特定导数公式，并将结果推广至矩阵舒伯特簇的任意并。
We consider the conormal bundle of a Schubert variety $S_I$ in the cotangent bundle $T^* Gr$ of the Grassmannian $Gr$ of $k$-planes in $C^n$. This conormal bundle has a fundamental class ${κ_I}$ in the equivariant cohomology $H^*_{T}(T^* Gr)$. Here $T=(C^*)^n\times C^*$. The torus $(C^*)^n$ acts on $T^* Gr$ in the standard way and the last factor $C^*$ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum $Y_I$ of the Yangian $Y(gl_2)$ weight functions $(W_J)_J$. We describe a relation of $Y_I$ with the double Schur polynomial $[S_I]$. A modified version of…
We use Young's raising operators to introduce and study double eta polynomials, which are an even orthogonal analogue of Wilson's double theta polynomials. Our double eta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of even orthogonal Grassmannians, and specialize to the single eta polynomials of Buch, Kresch, and the author.
Let $G$ be a classical complex Lie group, $P$ any parabolic subgroup of $G$, and $G/P$ the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in the cohomology ring of $G/P$ as a polynomial in certain special Schubert class generators. Our formula extends to one that applies to the torus-equivariant cohomology ring of $G/P$ and to the setting of symplectic and orthogonal degeneracy loci.
Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear …
A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. In particular, we construct a Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes using convex geometric push-pull operators in type A.
The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing $K$-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant $K$-classes of s…
Quantum Giambelli formulas for isotropic Grassmannians
设X是一个辛或奇正交格拉斯曼算子，它参数化了具有非退化（斜）对称形式的向量空间中的迷向子空间.我们证明了量子Giambelli公式，它将X的小量子上同调环中的任意Schubert类表示为某些特殊Schubert类中的多项式，推广了arXiv：0811.2781中的上同调Giambelli公式.
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。
Pieri-type formulas for maximal isotropic Grassmannains via triple intersections
给出了奇正交或辛向量空间的极大迷向子空间的Grassmannian的上同调中Pieri型公式的初等证明。这个证明的收益，明确计算一个三重交叉的舒伯特品种。决定性的一步是对两个舒伯特簇的交集的精确描述，从那里，皮耶里型公式中的多重性（是2的幂）是显而易见的。
Action of Weyl group on equivariant K-theory of flag varieties
本文研究半单线性群Weyl群对旗簇等变K-理论环的作用，利用分差算子和局部化定理计算反射自同构公式，并在Schubert簇结构层类基下展开，给出有效近似及motivic Chern类基下的讨论。
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…
Quantum cohomology of orthogonal Grassmannians
设V是具有非退化对称形式的向量空间，OG是V中极大迷向子空间的正交Grassmannian，给出了（小）量子上同调环QH^*（OG）的一个表示，并证明了它的乘积结构由（P~）-多项式环决定.一个“量子舒伯特演算”制定，其中包括量子皮耶里和Giambelli公式，以及计算Gromov-Witten不变量的算法。作为一个应用，我们表明，表的3点，属零Gromov-Witten不变量的OG相吻合，为相应的拉格朗日格拉斯曼LG，对合。
Symmetric polynomials and divided differences in formulas of intersection theory
本文的目的是双重的。首先，我们试图对对称多项式和差除法在交理论中的应用作一综述。我们讨论：多项式普遍支持的退化轨迹;一些明确的公式陈和Segre类Schur束与应用枚举几何;标志退化轨迹;基本类，对角线和Gysin地图;交叉环的G/P和公式的迷向退化轨迹;数值正多项式的充足的向量丛。本文除了综述已有的结果外，还包括一些新的结果和对以前结果的新证明：如何由周围空间的对角线类计算子簇的基本类，如何利用Gysin映射计算相对对角线类，Grassmannian丛中Schur Q-多项式的一个新的推进公式，Schur丛的全Chern类的一个新的公式，Schur丛的全Chern类的一个新的推广公式，Schur丛的全Chern类的一个新的推广公式，Schur丛的全Chern类的一个新的推广公式。另一个证明舒伯特和詹贝利的枚举完全二次;一个运营商证明雅可比特鲁迪公式;舒尔复杂证明詹贝利托姆波蒂厄斯公式。
Pieri-type formulas for maximal isotropic Grassmannians via triple intersections
给出了奇正交或辛向量空间的极大迷向子空间的格拉斯曼上同调的Pieri型公式的初等证明。这个证明的收益，明确计算一个三重交叉的舒伯特品种。决定性的一步是两个舒伯特簇的交集的明确描述，从中推导出Pieri型公式中的多重性（2的幂）。
Seidel and Pieri products in cominuscule quantum K-theory
本文研究了共最小旗形簇的量子 K-理论环中 Schubert 类的乘积公式，包括 Seidel 表示的 K-理论版本及新 Pieri 公式，以及对 Lagrangian 格拉姆-施密特流形的 Pieri 公式新证明。
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