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89
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 88 (7 self)
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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 Schubert polynomials. We also derive the quantum Monk's formula.
On the quantum product of Schubert classes
 Journal of Algebraic Geometry
"... Abstract. 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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Cited by 55 (3 self)
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Abstract. 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. 1.
Pieri’s formula for flag manifolds and Schubert polynomials, Ann
 MR MR1385512 (97g:14035
, 1996
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Schubert Polynomials, The Bruhat Order, And The Geometry Of Flag Manifolds
 Duke Math. J
, 1998
"... We illuminate the relation between the Bruhat order and structure constants for the polynomial ring in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We est ..."
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Cited by 47 (21 self)
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We illuminate the relation between the Bruhat order and structure constants for the polynomial ring in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.
Affine Weyl groups in Ktheory and representation theory
, 2004
"... We give an explicit combinatorial Chevalleytype formula for the equivariant Ktheory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is ..."
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Cited by 41 (22 self)
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We give an explicit combinatorial Chevalleytype formula for the equivariant Ktheory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is given in terms of a certain Rmatrix, that is, a collection of operators satisfying the YangBaxter equation. It reduces to combinatorics of decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. The formula implies several symmetries of coefficients in the equivariant Ktheory. We derive a Pieritype formula and a dual Chevalleytype formula for this ring. The paper contains some other applications and examples. Finally, we conjecture a Pieritype formula for the
Pattern Avoidance and Rational Smoothness of Schubert Varieties
 Adv. Math
, 1998
"... this article. Theorem 1.2. If Xw is a Schubert variety of type B or C, then the following conditions are equivalent: 1. Xw is rationally smooth ..."
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Cited by 39 (6 self)
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this article. Theorem 1.2. If Xw is a Schubert variety of type B or C, then the following conditions are equivalent: 1. Xw is rationally smooth
The Arason invariant and mod 2 algebraic cycles
 J. A.M.S
, 1998
"... 2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7 ..."
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Cited by 33 (8 self)
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2. The special Clifford group 5 3. Kcohomology of split reductive algebraic groups 7
Maximal singular loci of Schubert varieties on SL(n)/B
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, VOL. 355, NO. 10 (OCT., 2003), PP.; 39153945
, 2003
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