Schubert polynomials for the affine Grassmannian

Lam, Thomas

doi:10.1090/S0894-0347-06-00553-4

Schubert polynomials for the affine Grassmannian

Author: Thomas Lam
Journal: J. Amer. Math. Soc. 21 (2008), 259-281
MSC (2000): Primary 05E05; Secondary 14N15
DOI: https://doi.org/10.1090/S0894-0347-06-00553-4
Published electronically: October 18, 2006
MathSciNet review: 2350056
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the -Schur functions in homology and affine Schur functions in cohomology. The results are obtained by connecting earlier combinatorial work of ours to certain subalgebras of Kostant and Kumar's nilHecke ring and to work of Peterson on the homology of based loops on a compact group. As combinatorial corollaries, we settle a number of positivity conjectures concerning -Schur functions, affine Stanley symmetric functions and cylindric Schur functions.

1. Alberto Arabia, Cohomologie 𝑇-équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), no. 2, 129–165 (French, with English summary). MR 1015806
2. I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces 𝐺/𝑃, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933
3. Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, Equivariant homology and 𝐾-theory of affine Grassmannians and Toda lattices, Compos. Math. 141 (2005), no. 3, 746–768. MR 2135527, https://doi.org/10.1112/S0010437X04001228
4. Raoul Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35–61. MR 102803
5. Paul Edelman and Curtis Greene, Balanced tableaux, Adv. in Math. 63 (1987), no. 1, 42–99. MR 871081, https://doi.org/10.1016/0001-8708(87)90063-6
6. Sergey Fomin and Curtis Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), no. 1-3, 179–200. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661368, https://doi.org/10.1016/S0012-365X(98)00140-X
7. Sergey Fomin and Richard P. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196–207. MR 1265793, https://doi.org/10.1006/aima.1994.1009
8. D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math. 144 (2001), no. 2, 253–280. MR 1826370, https://doi.org/10.1007/s002220100122
9. Howard Garland and M. S. Raghunathan, A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), no. 12, 4716–4717. MR 417333, https://doi.org/10.1073/pnas.72.12.4716
10. V. GINZBURG: Perverse sheaves on a loop group and Langlands' duality, preprint; math.AG/9511007.
11. William Graham, Positivity in equivariant Schubert calculus, Duke Math. J. 109 (2001), no. 3, 599–614. MR 1853356, https://doi.org/10.1215/S0012-7094-01-10935-6
12. Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, https://doi.org/10.1090/S0894-0347-01-00373-3
13. James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
14. Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of 𝐺/𝑃 for a Kac-Moody group 𝐺, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 6, 1543–1545. MR 831908, https://doi.org/10.1073/pnas.83.6.1543
15. Bertram Kostant and Shrawan Kumar, 𝑇-equivariant 𝐾-theory of generalized flag varieties, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 13, 4351–4354. MR 894705, https://doi.org/10.1073/pnas.84.13.4351
16. Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198
17. T. LAM: Affine Stanley symmetric functions, Amer. J. Math., to appear; math.CO/0501335.
18. T. LAM: Schubert polynomials for the affine Grassmannian (extended abstract), Proc. FPSAC, 2006, San Diego.
19. T. LAM, L. LAPOINTE, J. MORSE, AND M. SHIMOZONO: Affine insertion and Pieri rules for the affine Grassmannian, preprint, 2006; math.CO/0609110.
20. T. LAM AND M. SHIMOZONO: A Little bijection for affine Stanley symmetric functions, preprint, 2006; math.CO/0601483.
21. L. Lapointe, A. Lascoux, and J. Morse, Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003), no. 1, 103–146. MR 1950481, https://doi.org/10.1215/S0012-7094-03-11614-2
22. L. Lapointe and J. Morse, Schur function analogs for a filtration of the symmetric function space, J. Combin. Theory Ser. A 101 (2003), no. 2, 191–224. MR 1961543, https://doi.org/10.1016/S0097-3165(02)00012-2
23. L. LAPOINTE AND J. MORSE: A -tableaux characterization of -Schur functions, preprint, 2005; math.CO/0505519.
24. L. LAPOINTE AND J. MORSE: Quantum cohomology and the -Schur basis, Tran. Amer. Math. Soc., to appear; math.CO/0501529.
25. Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
26. Alain Lascoux and Marcel-Paul Schützenberger, Schubert polynomials and the Littlewood-Richardson rule, Lett. Math. Phys. 10 (1985), no. 2-3, 111–124. MR 815233, https://doi.org/10.1007/BF00398147
27. I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598
28. D. PETERSON: Lecture notes at MIT, 1997.
29. Alexander Postnikov, Affine approach to quantum Schubert calculus, Duke Math. J. 128 (2005), no. 3, 473–509. MR 2145741, https://doi.org/10.1215/S0012-7094-04-12832-5
30. Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587