zbMATH — the first resource for mathematics

Combinatorics of the \(K\)-theory of affine grassmannians. (English) Zbl 1238.05276
Summary: We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and \(k\)-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and \(k-K\)-Schur functions – Schubert representatives for the \(K\)-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
17B65 Infinite-dimensional Lie (super)algebras
Full Text: DOI arXiv
[1] Bandlow, J.; Morse, J., Combinatorial expansions in k-theoretic bases · Zbl 1267.05037
[2] P. Berthelot, A. Grothendieck, L. Illusle, Theorie des intersections et theoreme de Riemann-Roch, Lecture notes from SGA VI (Seminaire de Geometrie Algebrique), 1971.
[3] Buch, A., A Littlewood-Richardson rule for the k-theory of Grassmannians, Acta math., 189, 36-78, (2002) · Zbl 1090.14015
[4] Demazure, M., Désingularization des variétés de Schubert, Annales E, 6, 53-88, (1974) · Zbl 0312.14009
[5] Fomin, S.; Greene, C., Non-commutative Schur functions and their applications, Discrete math., 193, 179-200, (1998) · Zbl 1011.05062
[6] Fomin, S.; Kirillov, A., Grothendieck polynomials and the Yang-Baxter equation, (), 183-190
[7] Kostant, B.; Kumar, S., The nil Hecke ring and cohomology of \(g / p\) for a Kac-Moody group g, Adv. math., 62, 187-237, (1986) · Zbl 0641.17008
[8] Kostant, B.; Kumar, S., t-equivariant k-theory of generalized flag varieties, J. differential geom., 549-603, (1990) · Zbl 0731.55005
[9] Lam, T., Affine Stanley symmetric functions, Amer. J. math., 128, 6, 1553-1586, (2006) · Zbl 1107.05095
[10] Lam, T., Schubert polynomials for the affine Grassmannian, J. amer. math. soc., 21, 1, 259-281, (2008) · Zbl 1149.05045
[11] T. Lam, L. Lapointe, J. Morse, M. Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc. (2009). · Zbl 1208.14002
[12] T. Lam, L. Lapointe, J. Morse, M. Shimozono, Poset on k-shapes and branching of k-Schur functions, Mem. Amer. Math. Soc. (2011). · Zbl 1292.05258
[13] T. Lam, P. Pylyavskyy, Combinatorial Hopf algebras and k-homology of Grassmannians, preprint. · Zbl 1134.16017
[14] T. Lam, A. Schilling, M. Shimozono, K-theory Schubert calculus of the affine Grassmannian, preprint. · Zbl 1256.14056
[15] Lapointe, L.; Lascoux, A.; Morse, J., Tableau atoms and a new Macdonald positivity conjecture, Duke math. J., 116, 1, 103-146, (2003) · Zbl 1020.05069
[16] Lapointe, L.; Morse, J., Tableaux on \(k + 1\)-cores, reduced words for affine permutations and k-Schur function expansions, J. combin. theory ser., 112, 44-81, (2005) · Zbl 1120.05093
[17] Lapointe, L.; Morse, J., A k-tableaux characterization for k-Schur functions, Adv. math., 183-204, (2007) · Zbl 1118.05096
[18] Lapointe, L.; Morse, J., Quantum cohomology and the k-Schur basis, Trans. amer. math. soc., 360, 2021-2040, (2008) · Zbl 1132.05060
[19] Lascoux, A., Puissances extérieures, déterminants et cycles de Schubert, Bull. soc. math. France, 102, 161-179, (1974) · Zbl 0295.14024
[20] Lascoux, A., Anneau de Grothendieck de la varieté de drapeaux, (), 1-34 · Zbl 0742.14041
[21] Lascoux, A., Symmetric functions and combinatorial operators on polynomials, CBMS notes, (2003), Amer. Math. Soc. Providence, RI · Zbl 1039.05066
[22] Lascoux, A.; Schützenberger, M.-P., Sur une conjecture de H.O. foulkes, C. R. acad. sci. Paris, 294, 323-324, (1978) · Zbl 0374.20010
[23] Lascoux, A.; Schützenberger, M.-P., Structure de Hopf de lʼanneau de cohomologie et de lʼanneau de Grothendieck dʼune variété de drapeaux, C. R. acad. sci. Paris Sér. I math., 629-633, (1982) · Zbl 0542.14030
[24] Lenart, C., Combinatorial aspects of the k-theory of Grassmannians, Ann. comb., 4, 67-82, (2000) · Zbl 0958.05128
[25] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Clarendon Press Oxford · Zbl 0487.20007
[26] M. Shimozono, M. Zabrocki, Stable Grothendieck polynomials and ω-calculus, preprint. · Zbl 0997.17017
[27] Stanley, R., Enumerative combinatorics, vol. 2, (1999), Cambridge University Press · Zbl 0928.05001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.