The Grassmannian Variety

Lakshmibai, V.; Brown, Justin

doi:10.1007/978-1-4939-3082-1

Cited by 14 publications

(6 citation statements)

References 59 publications

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“…In this section we fix notation and recall some definitions useful for the rest of the paper. We refer to [27] and [28] for posets and their incidence algebras, to [1] and [16] for the theory of Coxeter groups, to [2], [4] and [21] for matroids and flag matroids, to [7], [17], [18], and [26] for general results on Grassmannian and flag varieties.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

P-flag spaces and incidence stratifications

Bolognini,

Sentinelli

2019

Preprint

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For any finite poset P , we introduce a homogeneous space as a quotient of the general linear group with the incidence group of P . When P is a chain this quotient is a flag variety; for the trivial poset our construction gives a variety recently introduced in [22]. Moreover we provide decompositions for any set in a projective space, induced by the action of the incidence group of a suitable poset. In the classical cases of Grassmannians and flag varieties we recover, depending on the choice of the poset, the partition into Schubert cells and the matroid strata. Our general framework produces, for the homogeneous spaces corresponding to the trivial posets, a stratification by parking functions.

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Section: Notation and Preliminariesmentioning

confidence: 99%

P-flag spaces and incidence stratifications

Bolognini,

Sentinelli

2019

Preprint

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“…The smooth Schubert varieties in X are sub-grassmannians (see for example [22,Theorem 6.4.2] or [14, Theorem 1.1], and [3] or [31] for more details on the singular locus and the type of singularities). The partitions corresponding to these smooth Schubert varieties are of the form λ = (λ 1 , · · · , λ k ) with λ i = b for i ∈ [1, a] and λ i = 0 for i > a for some integers a ∈ [1, k] and b ∈ [1, n − k].…”

Section: Smooth Schubert Varieties Bott-samelson Resolution and Cobomentioning

confidence: 99%

“…See e.g. [13], [22] for more details. We give here an explicit description of these resolutions in the spirit of configuration spaces (see [28] or [30]).…”

Section: Smooth Schubert Varieties Bott-samelson Resolution and Cobomentioning

confidence: 99%

Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians

Hornbostel

Perrin

2018

Pacific J. Math.

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We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with any Bott-Samelson class in algebraic cobordism of the grassmannian. We also establish some results for generalized Schubert polynomials for hyperbolic formal group laws. * The second author was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

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“…In this section and the next we fix the notation that will be used throughout the paper. For a more in depth introduction to these topics see [LB15] and [LR08].…”

Section: Preliminariesmentioning

confidence: 99%

Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

Hodges

Lakshmibai

2017

Algebr Represent Theor

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Abstract. Let Lw be the Levi part of the stabilizer Qw in GLN (for left multiplication) of a Schubert variety X(w) in the Grassmannian G d,N . For the natural action of Lw on C[X(w)], the homogeneous coordinate ring of X(w) (for the Plücker embedding), we give a combinatorial description of the decomposition of C[X(w)] into irreducible Lw-modules; in fact, our description holds more generally for the action of the Levi part L of any parabolic subgroup Q that is contained in Qw. This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G2,N are spherical Lw-varieties.

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The Grassmannian Variety

Abstract: More information about this series at http://www.springer.com/series/5834

Cited by 14 publications

References 59 publications

P-flag spaces and incidence stratifications

P-flag spaces and incidence stratifications

Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians

Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

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