EQUIVARIANT $K$ -THEORY OF GRASSMANNIANS
Forum of Mathematics, Pi, Tome 5 (2017)

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We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$ -theory of Grassmannians, Acta Math. 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J.119(2) (2003), 221–260]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant $K$ -theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of Anderson et al. [Positivity and Kleiman transversality in equivariant $K$ -theory of homogeneous spaces, J. Eur. Math. Soc.13 (2011), 57–84] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of Schützenberger’s [Combinatoire et représentation du groupe symétrique, in Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113] jeu de taquin. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of Thomas and Yong [Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble) (2013), to appear]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.
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OLIVER PECHENIK; ALEXANDER YONG. EQUIVARIANT $K$ -THEORY OF GRASSMANNIANS. Forum of Mathematics, Pi, Tome 5 (2017). doi: 10.1017/fmp.2017.4

[AnGrMi11] Anderson, D., Griffeth, S. and Miller, E., ‘Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces’, J. Eur. Math. Soc. (JEMS) 13 (2011), 57–84.Google Scholar

[Br05] Brion, M., ‘Lectures on the geometry of flag varieties’, inTopics in Cohomological Studies of Algebraic Varieties, Trends in Mathematics (Birkhäuser, Basel, 2005), 33–85.Google Scholar

[Bu02] Buch, A., ‘A Littlewood–Richardson rule for the K-theory of Grassmannians’, Acta Math. 189 (2002), 37–78.Google Scholar

[Bu15] Buch, A., ‘Mutations of puzzles and equivariant cohomology of two-step flag varieties’, Ann. of Math. (2) 182 (2015), 173–220.Google Scholar

[BKPT16] Buch, A., 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

[BuKrTa03] Buch, A., Kresch, A. and Tamvakis, H., ‘Gromov–Witten invariants on Grassmannians’, J. Amer. Math. Soc. 16 (2003), 901–915.Google Scholar

[BuMi11] Buch, A. and Mihalcea, L., ‘Quantum K-theory of Grassmannians’, Duke Math. J. 156(3) (2011), 501–538.Google Scholar

[Co09] Coşkun, I., ‘A Littlewood–Richardson rule for two-step flag varieties’, Invent. Math. 176 (2009), 325–395.Google Scholar

[CoVa09] Coşkun, I. and Vakil, R., ‘Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus’, inAlgebraic Geometry—Seattle 2005, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 77–124, Part 1.Google Scholar

[FuLa94] Fulton, W. and Lascoux, A., ‘A Pieri formula in the Grothendieck ring of a flag bundle’, Duke Math. J. 76(3) (1994), 711–729.Google Scholar

[GiLe17] Gillespie, M. and Levinson, J., ‘Monodromy and K-theory of Schubert curves via generalized jeu de taquin’, J. Algebraic Combin. 45 (2017), 191–243.Google Scholar

[Gr01] Graham, W., ‘Positivity in equivariant Schubert calculus’, Duke Math. J. 109(3) (2001), 599–614.Google Scholar

[GrKu08] Graham, W. and Kumar, S., ‘On positivity in T-equivariant K-theory of flag varieties’, Int. Math. Res. Not. (2008), Art. ID rnn093, 43 pages.Google Scholar

[HLMvW11] Haglund, J., Luoto, K., Mason, S. and Van Willigenburg, S., ‘Refinements of the Littlewood–Richardson rule’, Trans. Amer. Math. Soc. 363 (2011), 1665–1686.Google Scholar

[Kn10] Knutson, A., ‘Puzzles, positroid varieties, and equivariant -theory of Grassmannians’, Preprint, 2010, arXiv:1008.4302.Google Scholar

[KnMiYo09] Knutson, A., Miller, E. and Yong, A., ‘Gröbner geometry of vertex decompositions and of flagged tableaux’, J. Reine Angew. Math. 630 (2009), 1–31.Google Scholar

[KnTa03] Knutson, A. and Tao, T., ‘Puzzles and (equivariant) cohomology of Grassmannians’, Duke Math. J. 119(2) (2003), 221–260.Google Scholar

[KoKu90] Kostant, B. and Kumar, S., ‘ T-equivariant K-theory of generalized flag varieties’, J. Differential Geom. 32 (1990), 549–603.Google Scholar

[Kr05] Kreiman, V., ‘Schubert classes in the equivariant -theory and equivariant cohomology of the Grassmannian’, Preprint, 2005, arXiv:math/0512204.Google Scholar

[LaSc82] 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

[LePo07] Lenart, C. and Postnikov, A., ‘Affine Weyl groups in K-theory and representation theory’, Int. Math. Res. Not. IMRN (12) (2007), Art. ID rnm038, 65 pp.Google Scholar

[LiRi34] Littlewood, D. E. and Richardson, A. R., ‘Group characters and algebra’, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 233(721–730) (1934), 99–141.Google Scholar

[MoSa99] Molev, A. and Sagan, B., ‘A Littlewood–Richardson rule for factorial Schur functions’, Trans. Amer. Math. Soc. 351(11) (1999), 4429–4443.Google Scholar

[PeYo15] Pechenik, O. and Yong, A., ‘Genomic tableaux and combinatorial K-theory’, Discrete Math. Theor. Comput. Sci. Proc. FPSAC‘15 (2015), 37–48. FPSAC 2015, Daejeon, Korea.Google Scholar

[PeYo17a] Pechenik, O. and Yong, A., ‘Equivariant K-theory of Grassmannians II: The Knutson–Vakil conjecture’, Compos. Math. 153 (2017), 667–677.Google Scholar

[PeYo17b] Pechenik, O. and Yong, A., ‘Genomic tableaux’, J. Algebraic Combin. 45 (2017), 649–685.Google Scholar

[RoYo15] Ross, C. and Yong, A., ‘Combinatorial rules for three bases of polynomials’, Sém. Lothar. Combin. 74 (2015), Art. B74a.Google Scholar

[Sc77] Schützenberger, M.-P., ‘Combinatoire et représentation du groupe symétrique’, inActes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113.Google Scholar

[ThYo09b] Thomas, H. and Yong, A., ‘A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus’, Algebra Number Theory 3(2) (2009), 121–148.Google Scholar

[ThYo13] Thomas, H. and Yong, A., ‘Equivariant Schubert calculus and jeu de taquin’, Ann. Inst. Fourier (Grenoble) (2013), to appear.Google Scholar

[WhZi16] Wheeler, M. and Zinn-Justin, P., ‘Littlewood–Richardson coefficients for Grothendieck polynomials from integrability’, Preprint, 2016, arXiv:1607.02396.Google Scholar

[Wi06] Willems, M., ‘ K-théorie équivariante des tours de Bott. Application à la structure multiplicative de la K-théorie équivariante des variétés de drapeaux’, Duke Math. J. 132(2) (2006), 271–309.Google Scholar

[WoYo12] Woo, A. and Yong, A., ‘A Gröbner basis for Kazhdan–Lusztig ideals’, Amer. J. Math. 134 (2012), 1089–1137.Google Scholar

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