Skew Schubert functions and the Pieri formula for flag manifolds

We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.

     

A partial Horn recursion in the cohomology of flag varieties

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.     

Double Schubert polynomials for the classical groups

For each infinite series of the classical Lie groups of type B, C or D, we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov.     

Schubert polynomials and classes of Hessenberg varieties

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 basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced.     

Chern class of Schubert cells in the flag manifold and related algebras

We discuss a relationship between Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds, the Fomin–Kirillov algebra, and the generalized nil-Hecke algebra. We show that the nonnegativity conjecture in the 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 Bott–Samelson varieties. We also discuss refined positivity conjectures of the Chern–Schwartz–MacPherson classes for Schubert cells motivated by the nonnegativity conjecture in the Fomin–Kirillov algebra.     

A Littlewood–Richardson rule for two-step flag varieties

This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties. As a consequence, we obtain a positive, geometric rule for expressing the structure constants of the cohomology of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for computing the structure constants of the small quantum cohomology of Grassmannians. We also obtain a positive, geometric rule for computing the classes of subvarieties of Grassmannians that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions. Mathematics Subject Classification (2000)  Primary 14M15, 14N35, 32M10     

Equivariant quantum Schubert polynomials

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

The skew Schubert polynomials

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen–Li–Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux–Pragacz identity for super-Schur functions. For the super-Lascoux–Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.     

Quantum Giambelli formulas for isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors?? cohomological Giambelli formulas.     

Schubert polynomials and Arakelov theory of orthogonal flag varieties

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety ${\mathfrak X={\rm SO}_N/B}$ . We use these polynomials to describe the arithmetic Schubert calculus on ${\mathfrak X}$ . Moreover, we give a method to compute the natural arithmetic Chern numbers on ${\mathfrak X}$ , and show that they are all rational numbers.     

From moment graphs to intersection cohomology

We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks. Received: 9 November 2000 / Published online: 24 September 2001     

The recursive nature of cominuscule Schubert calculus

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 whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.     

Quantum cohomology of flag manifolds

We study two aspects of quantum Schubert calculus: a presentation of the (small) quantum cohomology ring of partial flag manifolds and a quantum Giambelli formula. Our proof gives a relationship between universal Schubert polynomials as defined by Fulton and quantum Schubert polynomials, as defined by Fomin, Gelfand, and Postnikov, and later extended by Ciocan-Fontanine. Intersection theory on hyperquot schemes is an essential element of the proof.     

Schubert polynomials and Bott-Samelson varieties

Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.?The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott-Samelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.?This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux. Received: November 27, 1997     

A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.     

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.     

Springer fibers and Schubert points

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.