A partial Horn recursion in the cohomology of flag varieties

Richmond, Edward

doi:10.1007/s10801-008-0149-9

Cited by 14 publications

(13 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For further details, see [BeKu06]. We also learned much of the background from [Ri09]. Our formula uses the jeu de taquin introduced in [Sc77].…”

Section: Introductionmentioning

confidence: 99%

Root-theoretic Young diagrams and Schubert calculus II

Searles

2016

Journal of Combinatorics

View full text Add to dashboard Cite

ABSTRACT. We continue the study of root-theoretic Young diagrams (RYDs) from [SeYo13]. We provide an RYD formula for the GL n Belkale-Kumar product, after [KnPu11], and we give a translation of the indexing set of [BuKrTa09] for Schubert varieties of non-maximal isotropic Grassmannians into RYDs. We then use this translation to prove that the RYD formulas of [SeYo13] for Schubert calculus of the classical (co)adjoint varieties agree with the Pieri rules of [BuKrTa09], which were needed in the proofs of the (co)adjoint formulas.

show abstract

“…For further details, see [BeKu06]. We also learned much of the background from [Ri09]. Our formula uses the jeu de taquin introduced in [Sc77].…”

Section: Introductionmentioning

confidence: 99%

Root-theoretic Young diagrams and Schubert calculus II

Searles

2016

Journal of Combinatorics

View full text Add to dashboard Cite

show abstract

“…Our principal results are a combinatorial formula for the Belkale-Kumar structure constants, and using this formula, a way to factor each structure constant as a product of d 2 Littlewood-Richardson coefficients. 1 There are multiple known factorizations (such as in [Ri09]) into d − 1 factors, of which this provides a common refinement.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients

Knutson¹,

Purbhoo²

2011

Electron. J. Combin.

View full text Add to dashboard Cite

The Belkale-Kumar product on H * (G/P) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G = GL n , it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone.We show that for G/P a (d−1)-step flag manifold, each Belkale-Kumar structure constant is a product of d 2 Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into d − 1 factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants.These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward '04]; using this relation we give another proof of Ressayre's result.Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2. CONTENTS

show abstract

“…15 The proof of Theorem 1.15 is almost identical. Letλ,μ andν be three Schubert cycles in the Grassmannian G(a, 2n − a).…”

Section: Very Classical Gromov-witten Invariantsmentioning

confidence: 91%

“…An embedding of a product of flag varieties into another flag variety often leads to such relations (see [1,15] for related work). For example, let σ λ 1 , .…”

Section: Remark 119mentioning

confidence: 99%

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

Coskun

2009

Math. Ann.

View full text Add to dashboard Cite

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k 1 , . . . , k r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood-Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.

show abstract

A partial Horn recursion in the cohomology of flag varieties

Cited by 14 publications

References 10 publications

Root-theoretic Young diagrams and Schubert calculus II

Root-theoretic Young diagrams and Schubert calculus II

Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

Contact Info

Product

Resources

About