The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties

Carrell, James B.

doi:10.1090/pspum/056.1/1278700

Cited by 82 publications

(136 citation statements)

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“…In the case k D m, the equivalence of (a) and (b) specializes to a criterion, due to Carrell and Peterson [Car94], for rational smoothness of the Schubert variety X w .…”

mentioning

confidence: 99%

On the shape of Bruhat intervals

Björner

Ekedahl

2009

Ann. Math.

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Let .W; S / be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J Â S. Let W J denote the set of minimal coset representatives modulo the parabolic subgroup W J . For w 2 W J , let f w;J i denote the number of elements of length i below w in Bruhat order on W J (with notation simplified to f w i in the case when W J D W ). We show that 0 Ä i < j Ä`.w/ i implies f w;J i

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“…In the case k D m, the equivalence of (a) and (b) specializes to a criterion, due to Carrell and Peterson [Car94], for rational smoothness of the Schubert variety X w .…”

mentioning

confidence: 99%

On the shape of Bruhat intervals

Björner

Ekedahl

2009

Ann. Math.

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“…Indeed, D(v) has n 2 − ℓ(v) elements, and {(i, j) | s ji ≤ v} has n 2 − ℓ(v) elements whenever X v is smooth by a theorem of Carrell [Car94]. We believe this statement can be proved purely combinatorially.…”

Section: úLfarsson and Woomentioning

confidence: 82%

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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“…By a special case of a theorem of Carrell and Peterson ( [6]; see also [12], XII, §2), an affine Schubert variety is palindromic if and only if it is rationally smooth. This yields the corollary: Since we have enumerated all the palindromic Schubert varieties, the corollary can be proved ad hoc by checking that the singular ones satisfy rational Poincaré duality.…”

Section: Corollary 15 If G Is Not Of Type a There Are Only Finitelymentioning

confidence: 98%