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Equivariant $K$-theory of Grassmannians II: the Knutson–Vakil conjecture

Part of: Special varieties Algebraic combinatorics

Published online by Cambridge University Press:  09 March 2017

Oliver Pechenik  and
Alexander Yong
Oliver Pechenik
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA email [email protected]
Alexander Yong
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email [email protected]
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Abstract

In 2005, Knutson–Vakil conjectured a puzzle rule for equivariant $K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.

Keywords

Grassmannianequivariant K-theorypuzzleSchubert calculusLittlewood–Richardson rule

MSC classification

Primary: 14M15: Grassmannians, Schubert varieties, flag manifolds
Secondary: 05E05: Symmetric functions and generalizations 05E15: Combinatorial aspects of groups and algebras
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 4 , April 2017 , pp. 667 - 677
Copyright
© The Authors 2017 

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References

Anderson, D., Griffeth, S. and Miller, E., Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces , J. Eur. Math. Soc. 13 (2011), 57–84.Google Scholar
Buch, A., Mutations of puzzles and equivariant cohomology of two-step flag varieties , Ann. of Math. (2) 182 (2015), 173–220.CrossRefGoogle Scholar
Buch, A. S., Kresch, A., Purbhoo, K. and Tamvakis, H., The puzzle conjecture for the cohomology of two-step flag manifolds , J. Algebraic Combin. 44 (2016), 973–1007.Google Scholar
Buch, A., Kresch, A. and Tamvakis, H., Gromov–Witten invariants on Grassmannians , J. Amer. Math. Soc. 16 (2003), 901–915.Google Scholar
Coşkun, I. and Vakil, R., Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus , in Algebraic geometry—Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 77–124.Google Scholar
Fulton, W. and Lascoux, A., A Pieri formula in the Grothendieck ring of a flag bundle , Duke Math. J. 76 (1994), 711–729.Google Scholar
Knutson, A., Puzzles, positroid varieties, and equivariant K-theory of Grassmannians, Preprint (2010), arXiv:1008.4302.Google Scholar
Knutson, A. and Purbhoo, K., Product and puzzle formulae for GL(n) Belkale–Kumar coefficients , Electron. J. Combin. 18 (2011), P76.Google Scholar
Knutson, A. and Tao, T., Puzzles and (equivariant) cohomology of Grassmannians , Duke Math. J. 119 (2003), 221–260.Google Scholar
Knutson, A., Tao, T. and Woodward, C., The honeycomb model of GL n (ℂ) tensor products II: puzzles determine facets of the Littlewood–Richardson cone , J. Amer. Math. Soc. 17 (2004), 19–48.Google Scholar
Kreiman, V., Equivariant Littlewood–Richardson skew tableaux , Trans. Amer. Math. Soc. 362 (2010), 2589–2617.Google Scholar
Lascoux, A. and Schützenberger, M.-P., Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux , C. R. Acad. Sci. Paris 295 (1982), 629–633.Google Scholar
Pechenik, O. and Yong, A., Equivariant K-theory of Grassmannians, Preprint (2015), arXiv:1506.01992.Google Scholar
Purbhoo, K., Puzzles, tableaux, and mosaics , J. Algebraic Combin. 28 (2008), 461–480.CrossRefGoogle Scholar
Thomas, H. and Yong, A., Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble), to appear. Preprint (2012), arXiv:1207.3209.Google Scholar
Vakil, R., A geometric Littlewood–Richardson rule , Ann. of Math. (2) 164 (2006), 371–422.Google Scholar