Genomic tableaux

Pechenik, Oliver; Yong, Alexander

doi:10.1007/s10801-016-0720-8

Cited by 25 publications

(35 citation statements)

References 51 publications

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“…Another enriched cohomology theory is K-theory (i.e., the Grothendieck ring of algebraic vector bundles). There is a lattice rule [5] for the corresponding K-theoretic Littlewood-Richardson coefficients k ν λ,µ (another lattice rule uses genomic tableaux [22]). Question 1.…”

Section: Schubert Calculus and Complexity Theorymentioning

confidence: 99%

Vanishing of Littlewood–Richardson polynomials is in P

2019

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J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, andÉ. Tardos' algorithm for combinatorial linear programming.

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Section: Schubert Calculus and Complexity Theorymentioning

confidence: 99%

Vanishing of Littlewood–Richardson polynomials is in P

2019

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“…Pechenik and Yong [PY17] gives a combinatorial rule for calculating the K-theory Littlewood-Richardson coefficient c ν λµ by counting certain genomic tableaux of skew shape. We give an equivalent formulation (see Section 5.2) here in terms of circle tableaux.…”

Section: K-theoretic Puzzles and Tableauxmentioning

confidence: 99%

Puzzles in K-homology of Grassmannians

Pylyavskyy

Yang

2019

Pacific J. Math.

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Knutson, Tao, and Woodward [KTW04] formulated a Littlewood-Richardson rule for the cohomology ring of Grassmannians in terms of puzzles. Vakil [Vak06] and Wheeler-Zinn-Justin [WZ16] have found additional triangular puzzle pieces that allow one to express structure constants for K-theory of Grassmannians. Here we introduce two other puzzle pieces of hexagonal shape, each of which gives a Littlewood-Richardson rule for K-homology of Grassmannians. We also explore the corresponding eight versions of K-theoretic Littlewood-Richardson tableaux.

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“…A theme of modern Schubert calculus has been extending such theories to richer generalized cohomologies and in particular into the K-theory ring of algebraic vector bundles. For a partial survey of recent work related to K-theoretic Schubert calculus and the associated combinatorics, see [PY17].…”

Section: Interaction Between K-promotion and Deflationmentioning

confidence: 99%

Orbits of plane partitions of exceptional Lie type

Mandel

Pechenik

2018

European Journal of Combinatorics

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For each minuscule flag variety X, there is a corresponding minuscule poset, describing its Schubert decomposition. We study an action on plane partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass (1995). For plane partitions of height at most 2, D. Rush and X. Shi (2013) proved an instance of the cyclic sieving phenomenon, completely describing the orbit structure of this action. They noted their result does not extend to greater heights in general; however, when X is one of the two minuscule flag varieties of exceptional Lie type E, they conjectured explicit instances of cyclic sieving for all heights.We prove their conjecture in the case that X is the Cayley-Moufang plane of type E 6 . For the other exceptional minuscule flag variety, the Freudenthal variety of type E 7 , we establish their conjecture for heights at most 4, but show that it fails generally. We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case X is an even-dimensional quadric hypersurface. Our argument uses ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on plane partitions to combinatorics derived from K-theoretic Schubert calculus.

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Genomic tableaux

Cited by 25 publications

References 51 publications

Vanishing of Littlewood–Richardson polynomials is in P

Vanishing of Littlewood–Richardson polynomials is in P

Puzzles in K-homology of Grassmannians

Orbits of plane partitions of exceptional Lie type

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