Forest-Like Permutations

Bousquet-Mélou, Mireille; Butler, Steven

doi:10.1007/s00026-007-0322-1

Cited by 50 publications

(64 citation statements)

References 17 publications

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“…Recently, M. Bousquet-Mélou and S. Butler [7] have used this proposition to give a characterization of factorial Schubert varieties in terms of Bruhat-restricted pattern avoidance. This solves a conjecture that we had distributed during the preparation of this article.…”

Section: Proposition 2 the Schubert Variety X W Is Factorial If And mentioning

confidence: 93%

When is a Schubert variety Gorenstein?

Woo

Yong

2006

Advances in Mathematics

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A (normal) variety is Gorenstein if it is Cohen-Macaulay and its canonical sheaf is a line bundle. This property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro-geometric purposes. We give a combinatorial characterization of which Schubert varieties are Gorenstein, using in part a new generalization of pattern avoidance. We also give an explicit description as a line bundle of the canonical sheaf of a Gorenstein Schubert variety.

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Section: Proposition 2 the Schubert Variety X W Is Factorial If And mentioning

confidence: 93%

When is a Schubert variety Gorenstein?

Woo

Yong

2006

Advances in Mathematics

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“…(A proof of this formula appears in [BMB07].) Bousquet This question has motivation beyond mere curiosity.…”

Section: Questionsmentioning

confidence: 99%

“…Indeed, the weaker conditions of being factorial [BMB07] and of being Gorenstein [WY06] cannot be characterized by pattern avoidance since there exist examples where X v is not factorial (respectively not Gorenstein), w pattern contains v, and X w is factorial (respectively Gorenstein). Instead some more general form of pattern avoidance is required in the statement of those theorems.…”

Section: Questionsmentioning

confidence: 99%

Which Schubert varieties are local complete intersections?

Ulfarsson

Woo

2013

Proceedings of the London Mathematical Society

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Abstract. We characterize by pattern avoidance the Schubert varieties for GL n which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. Applications include formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties.

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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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Forest-Like Permutations

Cited by 50 publications

References 17 publications

When is a Schubert variety Gorenstein?

When is a Schubert variety Gorenstein?

Which Schubert varieties are local complete intersections?

Embedded Factor Patterns for Deodhar Elements in Kazhdan-Lusztig Theory

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