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101
Tensor product multiplicities, canonical bases and totally positive varieties
 Invent. Math
, 2001
"... 2. Tensor product multiplicities 4 2.1. Background on semisimple Lie algebras 4 2.2. Polyhedral expressions for tensor product multiplicities 4 ..."
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Cited by 110 (11 self)
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2. Tensor product multiplicities 4 2.1. Background on semisimple Lie algebras 4 2.2. Polyhedral expressions for tensor product multiplicities 4
KazhdanLusztig polynomials for 321hexagonavoiding permutations
 J. ALG. COMB
, 2001
"... In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the KazhdanLusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where W = Sn (the symmetric group on n letters) and the ..."
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Cited by 50 (5 self)
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In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the KazhdanLusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where W = Sn (the symmetric group on n letters) and the permutation w is 321hexagonavoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on KazhdanLusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1 + q) l(w) if and only if w is 321hexagonavoiding. We also give a sufficient condition for the Schubert variety Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of Xw when w is 321hexagonavoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (Bn, F4, G2).
Quasicommuting families of quantum Plücker coordinates
 Advances in Math. Sciences (Kirillov's seminar), AMS Translations 181
, 1998
"... this paper, we deal with the following problem ..."
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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
"... ..."
Lower bounds for KazhdanLusztig polynomials from patterns
 Transform. Groups
"... Abstract. Kazhdan–Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values Px,w(1) in terms of “patterns”. A pattern for an element of a Weyl group is its image under a ..."
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Cited by 26 (5 self)
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Abstract. Kazhdan–Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values Px,w(1) in terms of “patterns”. A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a onedimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for “hyperbolic localization ” [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties. 1.
Excited Young diagrams and equivariant Schubert calculus
 Trans. Amer. Math. Soc
"... Abstract. We describe the torusequivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ..."
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Cited by 24 (5 self)
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Abstract. We describe the torusequivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call “excited Young diagrams ” and the second one is written in terms of factorial Schur Q or Pfunctions. As an application, we give a Giambellitype formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety. 1.
Total positivity for cominuscule Grassmannians
 NEW YORK J. MATH.
, 2008
"... In this paper we explore the combinatorics of the nonnegative part (G/P)≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γdiagrams — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P)≥0. In the classical cases, we describe Γdiagrams ..."
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Cited by 23 (2 self)
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In this paper we explore the combinatorics of the nonnegative part (G/P)≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γdiagrams — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P)≥0. In the classical cases, we describe Γdiagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Γdiagram. We give enumerative results and relate our Γdiagrams to other combinatorial objects. Surprisingly, the totally nonnegative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference
Minuscule Elements of Weyl Groups
, 2000
"... This paper has two main objectives. First, it has been clear from the beginning of Proctor's work in [P1] that minuscule ..."
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Cited by 23 (2 self)
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This paper has two main objectives. First, it has been clear from the beginning of Proctor's work in [P1] that minuscule