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165
Koszul duality patterns in representation theory
 JOUR. AMER. MATH. SOC
, 1996
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A diagrammatic approach to categorification of quantum groups I
, 2009
"... To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph. ..."
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Cited by 182 (18 self)
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To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph.
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.
Schubert Polynomials for the Classical Groups
 J. Amer. Math. Soc
, 1994
"... Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special ..."
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Cited by 68 (5 self)
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Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modern solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of inter
Quadratic algebras, Dunkl elements, and Schubert calculus
 in Advances in Geometry (J.L. Brylinski
, 1997
"... We suggest a new combinatorial construction for the cohomology ring of the flag manifold. ..."
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Cited by 65 (10 self)
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We suggest a new combinatorial construction for the cohomology ring of the flag manifold.
Quantum cohomology of flag manifolds G/B and Toda lattices
"... Let G be a connected semisimple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B. ..."
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Cited by 65 (3 self)
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Let G be a connected semisimple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B.
Noncommutative Schur Functions and their Applications
"... We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many wellknown associative algebras. As an application of our theory, we prove Schurpositivity and obtain generalized LittlewoodRichardson and MurnaghanNakayama rules for a larg ..."
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Cited by 61 (1 self)
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We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many wellknown associative algebras. As an application of our theory, we prove Schurpositivity and obtain generalized LittlewoodRichardson and MurnaghanNakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials. 1. Introduction and Main Results In this paper we develop a theory of Schur functions in noncommuting variables, assuming certain commutation relations that are satisfied in many wellknown examples such as the plactic, nilplactic, and nilCoxeter algebras and the degenerate Hecke algebra Hn (0). We show that most of the classical theory of symmetric functions can be reproduced in this noncommutative setting. There are many combinatorial representations of these commutation relations, and to each of these one can associate a family of (ordinary) symmetric functions; examples of such families include skew ...
Schubert polynomials for the affine Grassmannian
 in preparation, 2005. POLYNOMIALS FOR THE AFFINE GRASSMANNIAN 13
"... Abstract. Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the kSchur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on th ..."
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Cited by 53 (14 self)
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Abstract. Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the kSchur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on noncommutative kSchur functions. 1.
Formulas For Lagrangian And Orthogonal Degeneracy Loci: The QPolynomials Approach
 COMPOSITIO MATH
, 1996
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ON QUANTUM COHOMOLOGY RINGS OF PARTIAL FLAG VARIETIES
 VOL. 98, NO. 3 DUKE MATHEMATICAL JOURNAL
, 1999
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