Criterion for smoothness of Schubert varieties in Sl(n)/B

Lakshmibai, V.; Sandhya, B

doi:10.1007/bf02881113

Cited by 171 publications

(171 citation statements)

References 3 publications

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“…3.11]. Similarly, permutations w ∈ S n for which the Schubert variety w in the complete flag variety GL(n, C)/B is smooth are those permutations that are 4231 and 3412-avoiding (implicit in Ryan [87], based on earlier work of Lakshmibai, Seshadri, and Deodhar, and explicit in Lakshmibai and Sandhya [69]). The enumeration of such "smooth permutations" in S n is due to Haiman [29], [52], [99,Exer.…”

Section: Pattern Avoidancementioning

confidence: 99%

Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. Mathematics Subject Classification (2000). Primary 05A05, 05A06; Secondary 60C05.

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Section: Pattern Avoidancementioning

confidence: 99%

Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…A good introduction to enumerative methods in pattern avoidance can be found in [Bón04]. Several interesting properties of Schubert varieties, Kazhdan-Lusztig polynomials and Bruhat order can be characterized by pattern avoidance [LS90,BP05,BB03,Gas98,WY06b,WY06a,Ten06a,BMB06].…”

Section: Heaps and String Diagramsmentioning

confidence: 99%

Embedded Factor Patterns for Deodhar Elements in Kazhdan-Lusztig Theory

Billey

Jones

2007

Ann. Comb.

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ABSTRACT. The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar [Deo90] has given a framework for computing the Kazhdan-Lusztig polynomials which generally involves recursion. We define embedded factor pattern avoidance for general Coxeter groups and use it to characterize when Deodhar's algorithm yields a simple combinatorial formula for the Kazhdan-Lusztig polynomials of finite Weyl groups. Equivalently, if (W, S) is a Coxeter system for a finite Weyl group, we classify the elements w ∈ W for which the Kazhdan-Lusztig basis element C ′ w can be written as a monomial of C ′ s where s ∈ S. This work generalizes results of Billey-Warrington [BW01] that identified the Deodhar elements in type A as 321-hexagon-avoiding permutations, and Fan-Green [FG97] that identified the fully-tight Coxeter groups.

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“…For example, Macdonald [8] (and others) have shown that the vexilliary permutations S n (2143) are relevant to the theory of Schubert polynomials, and therefore to the cohomological structure of flag manifolds; Knuth [5] has demonstrated that the permutations S n (231) are exactly those which are stack sortable; and a beautiful theorem of Lakshmibai and Sandhya [6] asserts that the permutations S n (1324, 2143) are those whose corresponding Schubert varieties are smooth. Recent work of Billey and Warrington [1] on (321)-hexagon-avoiding permutations, and of Mansour and Vainshtein [9] on S n (132), has developed relationships with Kashdan-Lusztig and Chebyshev polynomials respectively.…”

Section: Introduction and Notationmentioning

confidence: 99%

Posets of Matrices and Permutations with Forbidden Subsequences

Ray¹,

West

2003

Annals of Combinatorics

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Abstract. The enumeration of permutations with specific forbidden subsequences has applications in areas ranging from algebraic geometry to the study of sorting algorithms. We consider a ranked poset of permutation matrices whose global structure incorporates the solution to the equivalent problem of enumerating permutations which contain a required subsequence. We describe this structure completely for saturated chains of lengths one and two, so settling several new and general instances of the original problem, and conclude with a superficial asymptotic investigation of arbitrary chains whose length is small by comparison with the rank of its constituent permutations. The value of this approach is reflected in the appearance of closed polynomial formulae (related to the Robinson-Schensted correspondence) and of a framework for the systematic analysis of associated combinatorial questions; indeed, we begin by studying a simpler poset of 0-1 sequences as the natural environment in which to introduce our insertion and deletion operators.

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Criterion for smoothness of Schubert varieties in Sl(n)/B

Cited by 171 publications

References 3 publications

Increasing and decreasing subsequences and their variants

Increasing and decreasing subsequences and their variants

Embedded Factor Patterns for Deodhar Elements in Kazhdan-Lusztig Theory

Posets of Matrices and Permutations with Forbidden Subsequences

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