The Geometry of Flag Manifolds

Monk, Donald

doi:10.1112/plms/s3-9.2.253

Cited by 91 publications

(55 citation statements)

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“…Using geometry, Monk [28] and more generally Chevalley [7] established a formula for multiplication by linear Schubert polynomials (divisor Schubert classes). A Pieri-type formula for multiplication by an elementary or complete homogeneous symmetric polynomial (special Schubert class) was given in [22].…”

Section: Introductionmentioning

confidence: 99%

Skew Schubert functions and the Pieri formula for flag manifolds

Bergeron

Sottile

2001

Trans. Amer. Math. Soc.

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Abstract. We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.

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Section: Introductionmentioning

confidence: 99%

Skew Schubert functions and the Pieri formula for flag manifolds

Bergeron

Sottile

2001

Trans. Amer. Math. Soc.

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“…The following result, on which the main results of [2] and [3] are based, was first proved in [4] and a different proof was given in [2].…”

Section: Proof Here We Have Hmentioning

confidence: 99%

The classification of cohomology endomorphisms of certain flag manifolds

Duan¹,

Zhao²

2000

Pacific J. Math.

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“…It is classically known that the flag manifold, which parameterizes the set of projective flags, admits well-behaved topological decompositions, namely stratifications (in fact, cell decompositions); see [10,17]. Moreover, the classical Ehresmann-Bruhat order describes all the possible degenerations of a pair of flags in a linear space V under linear transformations of V .…”

Section: From Projective Real Flags To Affine Real Flagsmentioning

confidence: 99%

Stratifications of the Euclidean motion group with applications to robotics

et al. 2008

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In this paper we derive stratifications of the Euclidean motion group, which provide a complete description of the singular locus in the configuration space of a family of parallel manipulators, and we study the adjacency between the strata. We prove that classically known cell decompositions of the flag manifold restricted to the open subset parameterizing the affine real flags are still stratifications, and we introduce a refinement of the classical Ehresmann-Bruhat order that characterizes the adjacency between all the different strata. Then we show how, via a four-fold covering morphism, the stratifications of the Euclidean motion group are induced.

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The Geometry of Flag Manifolds

Cited by 91 publications

References 0 publications

Skew Schubert functions and the Pieri formula for flag manifolds

Skew Schubert functions and the Pieri formula for flag manifolds

The classification of cohomology endomorphisms of certain flag manifolds

Stratifications of the Euclidean motion group with applications to robotics

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