Schubert calculus
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Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians Open
In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian $\\mathbb{X}=\\mathit{Gr}_{n-k}(\\mathbb{C}^{n})$ , as well as t…
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The Positive Tropical Grassmannian, the Hypersimplex, and the<i>m</i>= 2 Amplituhedron Open
The positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroi…
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Degenerate Flag Varieties of Type A and C are Schubert Varieties Open
We show that any type A or C degenerate flag variety is, in fact, isomorphic to a Schubert variety in an appropriate partial flag manifold.
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Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds Open
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 …
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A Giambelli formula for isotropic Grassmannians Open
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…
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A Giambelli formula for even orthogonal Grassmannians Open
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 classic…
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EQUIVARIANT -THEORY OF GRASSMANNIANS Open
We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$ -theory of Grassmannians, Acta Math . 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grass…
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Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming Open
We establish connections between the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturm…
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A comparison of Newton–Okounkov polytopes of Schubert varieties Open
A Newton–Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (respectively, the first author and Naito) proved that the Newton–Okounkov body of a Schubert variety associated wit…
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Towards generalized cohmology Schubert calculus via formal root polynomials Open
An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work un…
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Notes on Schubert, Grothendieck and Key Polynomials Open
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properti…
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A Demazure crystal construction for Schubert polynomials Open
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur…
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Equivariant Schubert calculus and jeu de taquin Open
We introduce edge labeled Young tableaux. Our main results provide a corresponding analogue of Schützenberger’s theory of jeu de taquin . These are applied to the equivariant Schubert calculus of Grassmannians. Reinterpreting, we present n…
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Schubert puzzles and integrability I: invariant trilinear forms Open
The puzzle rules for computing Schubert calculus on $d$-step flag manifolds, proven in [Knutson Tao 2003] for $1$-step, in [Buch Kresch Purbhoo Tamvakis 2016] for $2$-step, and conjectured in [Coskun Vakil 2009] for $3$-step, lead to vecto…
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On the Singular Chern Classes of Schubert Varieties Via Small Resolution Open
We compute the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in a Grassmannian using a small resolution introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class using a small resolution instea…
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Positivity of Chern classes of Schubert cells and varieties Open
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.
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Totally nonnegative Grassmannian and Grassmann polytopes Open
These are lecture notes intended to supplement my second lecture at the Current Developments in Mathematics conference in 2014. In the first half of article, we give an introduction to the totally nonnegative Grassmannian together with a s…
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Schubert Calculus and the Homology of the Peterson Variety Open
We use the tight correlation between the geometry of the Peterson variety and the combinatorics the symmetric group to prove that homology of the Peterson variety injects into the homology of the flag variety. Our proof counts the points o…
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Whittaker functions, geometric crystals, and quantum Schubert calculus Open
This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an inte…
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Combinatorial models for Schubert polynomials Open
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other kno…
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Schubert presentation of the cohomology ring of flag manifolds Open
Let $G$ be a compact connected Lie group with a maximal torus $T$ . In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.
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Diagonal degenerations of matrix Schubert varieties Open
Knutson and Miller (2005) established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natura…
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Newton-Okounkov polytopes of Schubert varieties arising from cluster structures Open
The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this paper, we study Newton-Okounkov…
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A Schubert basis in equivariant elliptic cohomology Open
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, a…
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Coxeter combinatorics and spherical Schubert geometry Open
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
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Soergel calculus and Schubert calculus Open
We reduce some key calculations of compositions of morphisms between Soergel bimodules ("Soergel calculus") to calculations in the nil Hecke ring ("Schubert calculus"). This formula has several applications in modular representation theory.
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Maximal Newton Points and the Quantum Bruhat Graph Open
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
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Soergel Calculus and Schubert Calculus Open
We reduce some key calculations of compositions of morphisms between Soergel bimodules ("Soergel calculus") to calculations in the nil Hecke ring ("Schubert calculus").This formula has several applications in modular representation theory.
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The Grassmannian Variety: Geometric and Representation-Theoretic Aspects Open
Preface.- 1. Introduction.- Part I. Algebraic Geometry-A Brief Recollection - 2. Preliminary Material.- 3. Cohomology Theory.- 4. Grobner Bases.- Part II. Grassmannian and Schubert Varieties.- 5. The Grassmannian and Its Schubert Varieties…
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CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS Open
A Schubert variety in a rational homogeneous variety G/P is defined by the closure of an orbit of a Borel subgroup B of G. In general, Schubert varieties are singular, and it is an old problem to determine which Schubert varieties are smoo…