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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 74 (9 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
"... Abstract. 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) ..."
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Cited by 33 (3 self)
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Abstract. 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). Keywords: graph
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 ..."
Maximal singular loci of Schubert varieties on SL(n)/B
 Trans. Amer. Math. Soc
"... you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact inform ..."
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Cited by 19 (3 self)
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you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, noncommercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
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
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 17 (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.
Some Combinatorial Aspects of Reduced Words in Finite Coxeter Groups
 TRANS. AMER. MATH. SOC
, 1997
"... ..."
Monomials and TemperleyLieb Algebras
 J. Algebra
"... . We classify the "fully tight" simplylaced Coxeter groups, that is, the ones whose ijiavoiding KazhdanLusztig basis elements are monomials in the generators Bs i . We then investigate the basis of the TemperleyLieb algebra arising from the KazhdanLusztig basis of the associated Hecke algeb ..."
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Cited by 10 (5 self)
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. We classify the "fully tight" simplylaced Coxeter groups, that is, the ones whose ijiavoiding KazhdanLusztig basis elements are monomials in the generators Bs i . We then investigate the basis of the TemperleyLieb algebra arising from the KazhdanLusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis. 1. Introduction Let W be a Weyl group which we shall view as a Coxeter group with simple generators S. Every w 2 W may be written as a product s i 1 \Delta \Delta \Delta s i l where s i k 2 S. If l is minimal, we say this product is a reduced expression for w and we define the length of w to be `(w) = l. Let H be the Hecke algebra associated with the Weyl group W . We understand this to be the A := Z[v; v \Gamma1 ] algebra (where v is a square root of the indeterminate q associated with H) with basis fTw : w 2 Wg and multiplication satisfying (1) T 2 s = (q \Gamma 1)T s +q for s 2 S, and (2) TwTw 0 = Tww 0 w...