A Littlewood-Richardson rule for the K-theory of Grassmannians

Buch, Anders Skovsted

doi:10.1007/bf02392644

Cited by 218 publications

(336 citation statements)

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“…Then µ = ((6), (5, 4, 4, 3), (1)) and the partition ρ(µ) is (17, 16, 16, 15, 14, 14, 13, 6), which is illustrated below. In Example 1 of [4], the quiver coefficient c µ (r) was computed to be 1. Due to our different conventions, the corresponding term there is written 1⊗s ⊗s , which is indexed by the sequence of partitions (∅, (3,1), (1)).…”

Section: The Main Resultsmentioning

confidence: 99%

“…Each product of cycles on a given line is v j v −1 i , where the increasing subchain for that line is from v i to v j . (21,20,19,18,17,16,15,14,13,12,11) (19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9) (18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8) (16, 15, 14, 13, 12, 11, 10, 9, 8, 6, 5) (25, 24, 23, 22, 21)(20, 19)(17, 16)(15, 14, 13, 12, 11, 10, 9, 8, 6, 5, 4) (23, 22, 21, 19, 16, 14, 13, 12, 11, 10, 9, 8, 6, 5, 4, 3, 3, 2) (22,21,19,16,14,13,12,11,10,9,8,6,5,4,3,2,1) This chain corresponds to the following semistandard tableau of shape ρ. Remark 4.…”

Section: Alternative Formulas For Quiver Coefficientsmentioning

confidence: 99%

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Quiver coefficients are Schubert structure constants

Buch

Sottile

Yong

2005

Mathematical Research Letters

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Section: The Main Resultsmentioning

confidence: 99%

Section: Alternative Formulas For Quiver Coefficientsmentioning

confidence: 99%

Quiver coefficients are Schubert structure constants

Buch

Sottile

Yong

2005

Mathematical Research Letters

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“…Combinatorics often enters, through attempts to interpret positive quantities as enumerators, but it is by geometric means that the positivity is often first-or most easily-verified (a notable exception being the order of events relating [Buc02] and [Bri02]). In the typical setup, going back to Ehresmann [Ehr34], the cohomology ring in question possesses an additive basis of classes carried by algebraic subvarieties.…”

Section: Introductionmentioning

confidence: 99%

Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces

Anderson¹,

Griffeth²,

Miller³

2010

J. Eur. Math. Soc.

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Abstract. We prove the conjectures of and concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P . These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for nontransitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term-the top one-with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.

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“…We benefited from conversations with Anders Buch, Bill Fulton, Arun Ram and Chris Woodward. Most of our computational investigations were done using Anders Buch's package [4].…”

Section: Acknowledgmentsmentioning

confidence: 99%