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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 68 (7 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 37 (3 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
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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 21 (3 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.
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 17 (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
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 12 (0 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
Freely braided elements in Coxeter groups
"... Abstract. We introduce a notion of “freely braided element ” for simply laced Coxeter groups. We show that an arbitrary group element w has at most 2 N(w) commutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bou ..."
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Abstract. We introduce a notion of “freely braided element ” for simply laced Coxeter groups. We show that an arbitrary group element w has at most 2 N(w) commutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bound is achieved if w is freely braided. In the type A setting, we show that the bound is achieved only for freely braided w. 1.