W. Fulton, C. Woodward
We give a formula for the smallest powers of the quantum parameters q that occur in a product of Schubert classes in the (small) quantum cohomology of general flag varieties G/P. We also include a complete proof of Peterson's quantum version of Chevalley's formula, also for general G/P's.
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We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces G/P. While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that …
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…
In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [R, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.
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。
Multi-rigidity of Schubert classes in partial flag varieties
本文研究有理齐性空间中 Schubert 类的多重刚性，刻画 A、B、D 型部分旗簇情形，并由广义 Grassmann 簇及单群情形推出一般 $G/P$ 的刚性结论。
The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.
Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
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…
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.
Based on the multiplicative rule of Schubert classes obtained in [Du3], we present an algorithm computing the product of two arbitrary Schubert classes. As a result, the algorithm gives also a method to compute the integral cohomology ring of a flag manifold independent of the classical spectral sequence technique due to Leray and Borel.
Elliptic Schubert Classes and the Poincare Duality
本文通过 Kostant-Kumar 方法，证明了 Schubert 簇的椭圆类满足庞加莱对偶性。
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.
Under the assumption that the base field k has characteristic 0, we compute the algebraic cobordism fundamental classes of a family of Schubert varieties isomorphic to full and symplectic flag bundles. We use this computation to prove a formula for the push-forward class of the Bott-Samelson resolutions associated to the symplectic flag bundle. We specialise our formula to connective K-theory and the Grothendieck ring of vector bundles, hence obtaining a description for all Schubert classes.
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 $d$.
Let G be a compact connected Lie group and H, the centralizer of a one-parameter subgroup in G. Combining the ideas of Bott-Samelson resulotions of Schubert varieties and the enumerative formula on a twisted products of 2-spheres obatained in {Du2], we obtain a closed formula for multiplying Schubert classes in the flag manifold G/H.
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 suggest the point of view that the Schubert classes of the affine Grassmannian of a simple algebraic group should be considered as Schur-positive symmetric functions. In particular, we give a geometric explanation of the Schur positivity of k-Schur functions (at t = 1). We also put this in the context of the theory of combinatorial Hopf algebras.
We establish a Gysin formula for Schubert bundles and a strong version of the duality theorem in Schubert calculus on Grassmann bundles. We then combine them to compute the fundamental classes of Schubert bundles in Grassmann bundles, which yields a new proof of the Giambelli formula for vector bundles.
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Lagrangian Grassmannian. Our formulas rely on a result of Ghorpade-Raghavan, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Lagrangian Grassmannian.
Billey-Type Formula for KL-Schubert Classes in Hyperbolic Cohomology
本文研究KL-示性类在旗帜簇的K-理论和双曲上同调中的性质，推导出双曲上同调中Grassmannian上KL-示性类的Billey型公式。
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.
In 2006, Belkale and Kumar define a new product on the cohomology of flag varieties and use this new product to give an improved solution to the eigencone problem for complex reductive groups. In this paper, we give a generalization of the Belkale-Kumar product to the branching Schubert calculus setting. The study of Branching Schubert calculus attempts to understand the induced map on cohomology of an equivariant embedding of flag varieties. The main application of our work is a compact formulation of the solution to the branching eigencone problem.
We give positive formulas for the restriction of a Schubert Class to a T-fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Grobner degeneration of a Schubert variety in the neighborhood of a T-fixed point of the Grassmannian.
We consider the varieties of singular $m \times m$ complex matrices which may be either general, symmetric or skew-symmetric (with $m$ even). For these varieties we have shown in another paper that they had compact "model submanifolds", for the homotopy types of the Milnor fibers which are classical symmetric spaces in the sense of Cartan. In this paper we use these models, combined with results due to a number of authors concerning the Schubert decomposition of Lie groups, symmetric spaces via the Cartan model together with Iwasawa decomposition to give a cell decomposition of the global Miln…
Let V be a 2n-dimensional complex symplectic space. Let G' be the Lagrangian Grassmannian of maximal isotropic subspaces of V embedded via the inclusion i into the Grassmannian G of all n-dimensional subspaces of V. We discuss the restriction via i* of a Schubert class from H(G), as an integral linear combination of Schubert classes in H(G'). Among the main tools we mention Stembridge's results on shifted tableaux. Using these results and a generalization of the Macdonald-You identity from an earlier author's paper, we establish several related algebro-geometric formulas.
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…
Hodge rigidity of Chern classes
本文研究Schubert胞的Chern-Schwartz-MacPherson类的齐次分量，证明在合适条件下每个分量由不可约子簇表示，扩展Huh的结果并适用于更多情况。
We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.
We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B,C and D, and we find polynomial representatives for the Schubert classes in these rings. These representatives are given in terms of the same Pfaffian formulas which appear in the theory of factorial $P$- and $Q$-Schur functions. After specializing to equivariant cohomology, we interpret the resulting presentations and Pfaffian formulas in terms of Chern classes of tautological bundles.
We describe the integral cohomology rings of the flag manifolds of types B_n, D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.
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 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…
Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert ba…
The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.
Given two Schubert classes $σ_λ$ and $σ_μ$ in the quantum cohomology of a Grassmannian, we construct a partition $ν$, depending on $λ$ and $μ$, such that $σ_ν$ appears with coefficient 1 in the lowest (or highest) degree part of the quantum product $σ_λ\starσ_μ$. To do this, we show that for any two partitions $λ$ and $μ$, contained in a $k$-by-$(n-k)$ rectangle and such that the 180-degree rotation of one does not overlap the other, there is a third partition $ν$, also contained in the rectangle, such that the Littlewood-Richardson number $c_{λμ}^ν$ is 1.
In this paper, we prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in even infinitesimal cohomology theories of the Grassmannian bundle and the Lagrangian Grassmannian bundle.
Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results…
Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety $G/B$. For this first we need to twist the notion of elliptic characteristic class of Borisov-Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott-Samelson resolution of Schubert varieties we prove a BGG-type recursion for the elliptic classes, and study the Hecke algebra of our ellipti…
We address the problem of defining Schubert classes independently of a reduced word in equivariant elliptic cohomology, based on the Kazhdan-Lusztig basis of a corresponding Hecke algebra. We study some basic properties of these classes, and make two important conjectures about them: a positivity conjecture, and the agreement with the topologically defined Schubert classes in the smooth case. We prove some special cases of these conjectures.
In this paper, we study the T_w-equivariant cohomology of the weighted Grassmannians wGr(d,n) introduced by Corti-Reid where T_w is the n-dimensional torus that naturally acts on wGr(d,n). We introduce the equivariant weighted Schubert classes and, after we show that they form a basis of the equivariant cohomology, we give an explicit formula for the structure constants with respect to this Schubert basis. We also find a linearly independent subset {wu_1,...,wu_n} of Lie(T_w)^* such that those structure constants are polynomials in wu_i's with non-negative coefficients, up to a permutation on …
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。
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。
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive t…
We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'' and the second one is written in terms of factorial Schur $Q$- or $P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.
The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes who…
Multiplicity-free products of Schubert divisors
研究任意域上旗簇中舒伯特除子类无重数乘积在 Chow 环中的次数，能将其分解为舒伯特簇类线性组合，通过研究组合系数给出判别准则，确定系数情况。
We study the equivariant oriented cohomology ring $h_T(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott-Samelson classes in $h_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results by Brion, Knutson, Peterson, Tymoczko and others. We then focus on the equivariant oriented cohomology theory corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar's construction of the pa…
The hypergeometric solutions to the KZ equation contain a certain symmetric ``master function'', [SV]. Asymptotics of the solutions correspond to critical points of the master function and give Bethe vectors of the inhomogeneous Gaudin model, [RV]. The general conjecture is that the number of orbits of critical points equals the dimension of the relevant vector space, and that the Bethe vectors form a basis. In [ScV], a proof of the conjecture for the sl_2 KZ equation was given. The difficult part of the proof was to count the number of orbits of critical points of the master function. Here we…
In this paper we study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp bounds for when the Schubert classes fail to span and we describe the resulting geometry.
On the Classification of Schubert Varieties in Partial Flag Varieties
本文在部分旗流形中推广了Schubert流形的分类，通过根系折叠得到等价对，并确定了Schubert三重流形等价类的上界。
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…
We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formul…
For any complex reductive connected Lie group G, many of the structure constants of the ordinary cohomology ring H^*(G/B; Z) vanish in the Schubert basis, and the rest are strictly positive. We present a combinatorial game, the ``root game'', which provides some criteria for determining which of the Schubert intersection numbers vanish. The definition of the root game is manifestly invariant under automorphisms of G, and under permutations of the classes intersected. Although these criteria are not proven to cover all cases, in practice they work very well, giving a complete answer to the ques…
We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a non trivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the Gromov-Witten invariants, which turn out to be very symmetric.
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.
Combinatorial Aspects of Elliptic Schubert Calculus
本研究将旗流形舒伯特演算的两个组合结果推广到等变椭圆上同调，推导了椭圆舒伯特类的Billey型局部化公式与A型旗流形的pipe dream模型，还验证了其退化为K-理论时可还原经典公式。
Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $λ=(λ_1,...,λ_k)$ with $λ_1\leq n$ there is a Schubert variety $X(λ).$ Let $T$ denote a maximal torus of the symplectic group acting on $LG_n.$ Consider the $T$-equivariant cohomology of $LG_n$ and the $T$-equivariant fundamental class $σ(λ)$ of $X(λ).$ The main result of the present paper is an explicit formula for the restriction of the class $σ(λ)$ to any torus fixed point. The formula is written in terms of fact…
The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph. We define path Schubert polynomials, which are quantum cohomology analogues of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quant…
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。
We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we p…
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