Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T04:43:33.676Z Has data issue: false hasContentIssue false
  • English
  • Français

K-theory Schubert calculus of the affine Grassmannian

Part of: Algebraic combinatorics Projective and enumerative geometry

Published online by Cambridge University Press:  26 January 2010

Thomas Lam ,
Anne Schilling  and
Mark Shimozono
Thomas Lam
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: [email protected])
Anne Schilling
Affiliation:
Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA (email: [email protected])
Mark Shimozono
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

Keywords

affine GrassmannianK-theorySchubert calculussymmetric functionsGKM condition

MSC classification

Secondary: 05E05: Symmetric functions and generalizations 14N15: Classical problems, Schubert calculus
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 811 - 852
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Andersen, H. H., Jantzen, J. C. and Soergel, W., Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque 220 (1994), 321.Google Scholar
[2]Bandlow, J. and Morse, J., Work in progress.Google Scholar
[3]Bezrukavnikov, R., Finkelberg, M. and Mirković, I., Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compositio Math. 141 (2005), 746–768.CrossRefGoogle Scholar
[4]Billey, S., Kostant polynomials and the cohomology ring for G/B, Duke Math. J. 96 (1999), 205–224.CrossRefGoogle Scholar
[5]Buch, A., A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37–78.CrossRefGoogle Scholar
[6]Chriss, N. and Ginzburg, V., Representation theory and complex geometry (Birkhäuser, Boston, MA, 1997).Google Scholar
[7]Demazure, M., Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I).CrossRefGoogle Scholar
[8]Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), 187–198.CrossRefGoogle Scholar
[9]Fomin, S. and Kirillov, A., Grothendieck polynomials and the Yang–Baxter equation, in Proc. sixth int. conf. on formal power series and algebraic combinatorics (DIMACS, Piscataway, NJ, 1994), 183–190.Google Scholar
[10]Goresky, M., Kottwitz, R. and MacPherson, R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83.CrossRefGoogle Scholar
[11]Goresky, M., Kottwitz, R. and MacPherson, R., Homology of affine Springer fibers in the unramified case, Duke Math. J. 121 (2004), 509–561.CrossRefGoogle Scholar
[12]Graham, W., Equivariant K-theory and Schubert varieties, Preprint (2002).Google Scholar
[13]Harada, M., Henriques, A. and Holm, T., Computation of generalized equivariant cohomologies of Kac–Moody flag varieties, Adv. Math. 197 (2005), 198–221.CrossRefGoogle Scholar
[14]Hivert, F. and Thiéry, N. M., MuPAD-Combinat, an open-source package for research in algebraic combinatorics, Séminaire Lotharingien de Combinatoire 51 (2003), B51zhttp://mupad-combinat.sourceforge.net/.Google Scholar
[15]Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[16]Kac, V. G., Infinite-dimensional Lie algebras, third edition (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[17]Kashiwara, M., The flag manifold of Kac–Moody Lie algebra, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) (Johns Hopkins University Press, Baltimore, MD, 1989), 161–190.Google Scholar
[18]Kashiwara, M. and Shimozono, M., Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials, Duke Math. J. 148 (2009), 501–538.CrossRefGoogle Scholar
[19]Kostant, B. and Kumar, S., The nil Hecke ring and cohomology of G/P for a Kac–Moody group G, Adv. Math. 62 (1986), 187–237.CrossRefGoogle Scholar
[20]Kostant, B. and Kumar, S., T-equivariant K-theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549–603.CrossRefGoogle Scholar
[21]Kumar, S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204 (Birkhäuser, Boston, MA, 2002).CrossRefGoogle Scholar
[22]Lam, T., Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), 1553–1586.CrossRefGoogle Scholar
[23]Lam, T., Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc. 21 (2008), 259–281.CrossRefGoogle Scholar
[24]Lam, T., Lapointe, L., Morse, J. and Shimozono, M., Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc., to appear, arXiv:math.CO/0609110.Google Scholar
[25]Lam, T. and Pylyavskyy, P., Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not. 2007 (2007), rnm 125.Google Scholar
[26]Lapointe, L., Lascoux, A. and Morse, J., Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003), 103–146.CrossRefGoogle Scholar
[27]Lapointe, L. and Morse, J., Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions, J. Combin. Theory Ser. A 112 (2005), 44–81.CrossRefGoogle Scholar
[28]Lapointe, L. and Morse, J., A k-tableau characterization of k-Schur functions, Adv. Math. 213 (2007), 183–204.CrossRefGoogle Scholar
[29]Lenart, C., Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb. 4 (2000), 67–82.CrossRefGoogle Scholar
[30]Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, second edition (Oxford University Press, New York, 1995).CrossRefGoogle Scholar
[31]Magyar, P., Notes on Schubert classes of a loop group, Preprint, arXiv:0705.3826.Google Scholar
[32]Morse, J., Combinatorics of the K-theory of affine Grassmannians, Preprint, arXiv:0907.0044.Google Scholar
[33]Peterson, D., Lecture notes at MIT (1997).Google Scholar
[34]Pressley, A. and Segal, G., Loop groups, Oxford Mathematical Monographs (Oxford University Press, Oxford, 1986).Google Scholar
[35]Sage, , Open Source Mathematics Software, http://www.sagemath.org/, and *-Combinat, http://wiki.sagemath.org/combinat.Google Scholar
[36]Stanley, R., Enumerative combinatorics: Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 2001).Google Scholar
[37]Willems, M., Cohomologie et K-théorie équivariantes des variétés de Bott-Samelson et des variétés de drapeaux, Bull. Soc. Math. France 132 (2004), 569–589.CrossRefGoogle Scholar