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Transformation Groups

, Volume 22, Issue 4, pp 1041–1079 | Cite as

A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

  • C. LENART
  • S. NAITO
  • D. SAGAKI
  • A. SCHILLINGEmail author
  • M. SHIMOZONO
Article

Abstract

We establish the equality of the specialization E (x ; q; 0) of the nonsymmetric Macdonald polynomial E (x ; q; t) at t = 0 with the graded character gch U w + (λ) of a certain Demazure-type submodule U w + (λ) of a tensor product of “single-column” Kirillov–Reshetikhin modules for an untwisted affine Lie algebra, where λ is a dominant integral weight and w is a (finite) Weyl group element; this generalizes our previous result, that is, the equality between the specialization P λ(x ; q; 0) of the symmetric Macdonald polynomial P λ(x ; q; t) at t = 0 and the graded character of a tensor product of single-column Kirillov–Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of nonsymmetric Macdonald polynomials: one in terms of quantum Lakshmibai–Seshadri paths and the other in terms of the quantum alcove model.

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References

  1. [BN]
    J. Beck, H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), 335–402.zbMATHMathSciNetGoogle Scholar
  2. [BB]
    A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics Vol. 231, Springer, New York, 2005.Google Scholar
  3. [I]
    B. Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), 299–318.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [INS]
    M. Ishii, S. Naito, D. Sagaki, Semi-infinite LakshmibaiSeshadri path model for level-zero extremal weight modules over quantum affine algebras, Adv. Math. 290 (2016), 967–1009.Google Scholar
  5. [Kac]
    V. G. Kac, Infinite Dimensional Lie Algebras, 3rd Edition, Cambridge University Press, Cambridge, 1990.Google Scholar
  6. [Kas1]
    M. Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), 383–413.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Kas2]
    M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), 117–175.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Kas3]
    M. Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41 (2005), 223–250.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [LS]
    T. Lam, M. Shimozono, Quantum cohomology of G/P and homology of affine Grassmannian, Acta Math. 204 (2010), 49–90.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [L]
    C. Lenart, On the combinatorics of crystal graphs, I. Lusztig's involution, Adv. Math. 211 (2007), 324–340.Google Scholar
  11. [LL1]
    C. Lenart, A. Lubovsky, A generalization of the alcove model and its applications, J. Algebraic Comb. 41 (2015), 751–783.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [LL2]
    C. Lenart, A. Lubovsky, A uniform realization of the combinatorial R-matrix, preprint 2015, arXiv:1503.01765.Google Scholar
  13. [LNS31]
    C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, A uniform model for KirillovReshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, Int. Math. Res. Not. 2015 (2015), 1848–1901.Google Scholar
  14. [LNS32]
    C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, A uniform model for KirillovReshetikhin crystals II: Alcove model, path model, and P = X, Int. Math. Res. Not. 2016, doi:  10.1093/imrn/rnw129.
  15. [LNS33]
    C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, Quantum LakshmibaiSeshadri paths and root operators, in: Proceedings of the 5th Mathematical Society of Japan Seasonal Institute: Schubert Calculus, Osaka, Japan, 2012, Advanced Studies in Pure Mathematics 71 (2016), 267–294.Google Scholar
  16. [LP1]
    C. Lenart, A. Postnikov, Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. 2007 (2007), 1–65.zbMATHGoogle Scholar
  17. [LP2]
    C. Lenart, A. Postnikov, A combinatorial model for crystals of Kac-Moody algebras, Trans. Amer. Math. Soc. 360 (2008), 4349–4381.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [L1]
    P. Littelmann, A LittlewoodRichardson rule for symmetrizable KacMoody algebras, Invent. Math. 116 (1994), 329–346.Google Scholar
  19. [L2]
    P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), 499–525.Google Scholar
  20. [M]
    I. G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.Google Scholar
  21. [NS1]
    S. Naito, D. Sagaki, LakshmibaiSeshadri paths of a level-zero weight shape and one-dimensional sums associated to level-zero fundamental representations, Compos. Math. 144 (2008), 1525–1556.Google Scholar
  22. [NS2]
    S. Naito, D. Sagaki, Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials, Math. Zeit. 283 (2016), 937–978.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [OS]
    D. Orr, M. Shimozono, Specializations of nonsymmetric MacdonaldKoornwinder polynomials, preprint 2013, arXiv:1310.0279v1.Google Scholar
  24. [P]
    D. Peterson, Quantum cohomology of G/P, Lecture Notes, Massachusetts Institute of Technology, Cambridge, MA, Spring 1997.Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • C. LENART
    • 1
  • S. NAITO
    • 2
  • D. SAGAKI
    • 3
  • A. SCHILLING
    • 4
    Email author
  • M. SHIMOZONO
    • 5
  1. 1.Departament of Mathematics and StatisticsState University of New York at AlbanyAlbanyUSA
  2. 2.Department of MathematicsTokyo Institute of TechnologyTokyoJapan
  3. 3.Institute of MathematicsUniversity of TsukubaTsukubaJapan
  4. 4.Department of MathematicsUniversity of CaliforniaDavisUSA
  5. 5.Department of Mathematics, MC 0151, 460 McBryde HallVirginia TechBlacksburgUSA

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