Small resolutions of singularities of Schubert varieties

Zelevinskii, A. V.

doi:10.1007/bf01083143

Cited by 71 publications

(83 citation statements)

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“…This lemma is essentially proved both in [Zel83] and [Per07, Sect. 5.4], and we leave the details to the reader.…”

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

“…Zelevinsky [Zel83] shows that the image of π P is indeed X w . Furthermore, given any ordering P of the peaks, there is a map ρ P : Z w → Z P from the BottSamelson variety (for our fixed choice of w) to the Zelevinsky variety Z P , defined by…”

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

“…The Zelevinsky resolution. Our description of the Zelevinsky resolution is based on the original one of Zelevinsky [Zel83], though following Perrin [Per07, Sect. 5], we unwind some of the inductive definitions.…”

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

“…Our second construction is based on a geometric interpretation of the Lascoux-Schützenberger formula due to Zelevinsky [Zel83]. He constructs some resolutions of singularities for Schubert varieties and shows geometrically that Kazhdan-Lusztig polynomials can be calculated from a cell decomposition (in the usual topological sense) of this resolution.…”

Section: Introductionmentioning

confidence: 99%

“…First we describe the resolutions of Zelevinsky [Zel83] and a map from Bott-Samelson resolutions to Zelevinsky resolutions. The main result in this section is Theorem 6.14, which gives our second construction realizing the Lascoux-Schützenberger formula in Deodhar's framework.…”

Section: Introductionmentioning

confidence: 99%

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Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

Jones

Woo

2012

Ann. Comb.

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ABSTRACT. We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schützenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.

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“…This lemma is essentially proved both in [Zel83] and [Per07, Sect. 5.4], and we leave the details to the reader.…”

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

Section: Cells Of the Bott-samelson Varietymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

Jones

Woo

2012

Ann. Comb.

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Bioriented flags and resolutions of Schubert varieties

Cibotaru

2020

Mathematische Nachrichten

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We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W‐flag varieties, algebro‐geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.

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