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Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 102 (15 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix
Governing singularities of Schubert varieties
 J. Algebra
"... ABSTRACT. We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable ” invariants P of singularities, we geometrically prove that th ..."
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Cited by 13 (7 self)
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ABSTRACT. We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable ” invariants P of singularities, we geometrically prove
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 176 (3 self)
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in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
Quantum Schubert Polynomials
 J. AMER. MATH. SOC
, 1997
"... We compute GromovWitten invariants of the flag manifold using a new combinatorial construction for its quantum cohomology ring. Our construction provides quantum analogues of the BernsteinGelfandGelfand results on the cohomology of the flag manifold, and the LascouxSchutzenberger theory of S ..."
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Cited by 86 (7 self)
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of Schubert polynomials. We also derive the quantum Monk's formula.
Schubert Polynomials and BottSamelson Varieties
 Comment. Math. Helv
, 1998
"... 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 express ..."
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Cited by 10 (1 self)
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expressions for Schubert polynomials. The above results arise naturally from a new geometric model of Schubert polynomials in terms of BottSamelson varieties. Our analysis includes a new, explicit construction for a BottSamelson variety Z as the closure of a Borbit in a product of flag varieties
SCHUBERT POLYNOMIALS AND CLASSES OF HESSENBERG VARIETIES
, 710
"... Abstract. 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 var ..."
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Cited by 1 (0 self)
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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
Schubert polynomials and quiver formulas
 Duke Math. J
, 2003
"... Abstract. Fulton’s universal Schubert polynomials [F3] represent degeneracy loci for morphisms of vector bundles with rank conditions coming from a permutation. The quiver formula of BuchFulton [BF] expresses these polynomials as an integer linear combination of products of Schur determinants. We p ..."
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Cited by 26 (15 self)
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present a positive, nonrecursive combinatorial formula for the coefficients. Our result is applied to obtain new expansions for the Schubert polynomials of Lascoux and Schützenberger [LS1] and explicit Giambelli formulas in the classical and quantum cohomology ring of any partial flag variety. 1.
Skew Schubert Polynomials
 Proc. Amer. Math. Soc
"... We de ne skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a ag manifold. We show that this de nition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in ..."
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Cited by 9 (4 self)
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in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coecients that are precisely the structure constants of the cohomology of the complex ag variety with respect to its basis of Schubert classes. We rederive
Traffic and related selfdriven manyparticle systems
, 2000
"... Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? ..."
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Cited by 336 (38 self)
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Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? What are the mechanisms behind stopandgo traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ‘‘freeze by heating’’? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to selfdriven manyparticle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particlebased), mesoscopic (gaskinetic), and macroscopic (fluiddynamic) models. Attention is also paid to the formulation of a micromacro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for selfdriven manyparticle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socioeconomic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.
Universal Schubert Polynomials
 Duke Math. J
, 1997
"... this paper is to introduce some polynomials that specialize to all previously known Schubert polynomials: the classical Schubert polynomials of Lascoux and Schutzenberger [LS], [M], the quantum Schubert polynomials of Fomin, Gelfand, and Postnikov [FGP], and quantum Schubert polynomials for parti ..."
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Cited by 14 (0 self)
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for partial flag varieties of CiocanFontanine [CF2]. There are also double versions of these universal Schubert polynomials that generalize the previously known double Schubert polynomials [LS], [M], [KM], [CFF]. They describe degeneracy loci of maps of vector bundles, but in a more general setting than
Results 1  10
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