Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Specialization of nonsymmetric Macdonald polynomials at $ t=\infty$ and Demazure submodules of level-zero extremal weight modules


Authors: Satoshi Naito, Fumihiko Nomoto and Daisuke Sagaki
Journal: Trans. Amer. Math. Soc. 370 (2018), 2739-2783
MSC (2010): Primary 17B37; Secondary 33D52, 81R60, 81R50
DOI: https://doi.org/10.1090/tran/7114
Published electronically: November 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give a representation-theoretic interpretation of the specialization $ E_{w_\circ \lambda } (q,\infty )$ of the nonsymmetric Macdonald polynomial $ E_{w_\circ \lambda }(q,t)$ at $ t=\infty $ in terms of the Demazure submodule $ V_{w_\circ }^- (\lambda )$ of the level-zero extremal weight module $ V(\lambda )$ over a quantum affine algebra of an arbitrary untwisted type. Here, $ \lambda $ is a dominant integral weight, and $ w_\circ $ denotes the longest element in the finite Weyl group $ W$. Also, for each $ x \in W$, we obtain a combinatorial formula for the specialization $ E_{x \lambda } (q, \infty )$ at $ t=\infty $ of the nonsymmetric Macdonald polynomial $ E_{x \lambda } (q,t)$ and also a combinatorical formula for the graded character $ \mathop {\rm gch}\nolimits V_{x}^- (\lambda )$ of the Demazure submodule $ V_{x}^- (\lambda )$ of $ V(\lambda )$. Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape $ \lambda $.


References [Enhancements On Off] (What's this?)

  • [BB] Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
  • [BFP] Francesco Brenti, Sergey Fomin, and Alexander Postnikov, Mixed Bruhat operators and Yang-Baxter equations for Weyl groups, Internat. Math. Res. Notices 8 (1999), 419-441. MR 1687323, https://doi.org/10.1155/S1073792899000215
  • [BN] Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335-402. MR 2066942
  • [Ch1] Ivan Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Internat. Math. Res. Notices 9 (1992), 171-180. MR 1185831, https://doi.org/10.1155/S1073792892000199
  • [Ch2] Ivan Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices 10 (1995), 483-515. MR 1358032, https://doi.org/10.1155/S1073792895000341
  • [CO] Ivan Cherednik and Daniel Orr, Nonsymmetric difference Whittaker functions, Math. Z. 279 (2015), no. 3-4, 879-938. MR 3318254, https://doi.org/10.1007/s00209-014-1397-0
  • [FM] E. Feigin and I. Makedonskyi, Generalized Weyl modules, alcove paths and Macdonald polynomials, Selecta. Math. (N. S.) 23 (2017), 2863-2897, DOI 10.1007/s00029-017-0346-2.
  • [HK] Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971
  • [I] Bogdan Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), no. 2, 299-318. MR 1953294, https://doi.org/10.1215/S0012-7094-03-11624-5
  • [INS] Motohiro Ishii, Satoshi Naito, and Daisuke Sagaki, Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras, Adv. Math. 290 (2016), 967-1009. MR 3451944, https://doi.org/10.1016/j.aim.2015.11.037
  • [Kac] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [Kas1] Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383-413. MR 1262212, https://doi.org/10.1215/S0012-7094-94-07317-1
  • [Kas2] Masaki Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117-175. MR 1890649, https://doi.org/10.1215/S0012-9074-02-11214-9
  • [Kas3] Masaki Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 223-250. MR 2115972
  • [Kat] S. Kato, Demazure character formula for semi-infinite flag manifolds, preprint 2016, arXiv:1605.04953.
  • [L] Peter Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3, 329-346. MR 1253196, https://doi.org/10.1007/BF01231564
  • [LNSSS1] Cristian Lenart, Satoshi Naito, Daisuke Sagaki, Anne Schilling, and Mark Shimozono, A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph, Int. Math. Res. Not. IMRN 7 (2015), 1848-1901. MR 3335235, https://doi.org/10.1093/imrn/rnt263
  • [LNSSS2] C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals II: Alcove model, path model, and P = X, Int. Math. Res. Not. IMRN 14 (2017), 4259-4319, doi:10.1093/imrn/mw129.
  • [LNSSS3] C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at t = 0 and level- zero Demazure characters, to appear in Transform. Groups, DOI 10.1007/s00031-017-9421-1.
  • [LNSSS4] C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, Quantum Lakshmibai-Seshadri paths and root operators, Adv. Stud. Pure Math. 71 (2016), 267-294.
  • [LS] Thomas Lam and Mark Shimozono, Quantum cohomology of $ G/P$ and homology of affine Grassmannian, Acta Math. 204 (2010), no. 1, 49-90. MR 2600433, https://doi.org/10.1007/s11511-010-0045-8
  • [M] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. MR 1976581
  • [M1] I. G. Macdonald, A new class of symmetric functions, Publ. I.R.M.A., Strasbourg, Actes 20-e Seminaire Lotharingen, 1988, pp. 131-171.
  • [NS1] Satoshi Naito and Daisuke Sagaki, Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra, Int. Math. Res. Not. 14 (2005), 815-840. MR 2146858, https://doi.org/10.1155/IMRN.2005.815
  • [NS2] Satoshi Naito and Daisuke Sagaki, Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations, Compos. Math. 144 (2008), no. 6, 1525-1556. MR 2474320, https://doi.org/10.1112/S0010437X08003606
  • [NS3] Satoshi Naito and Daisuke Sagaki, Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials, Math. Z. 283 (2016), no. 3-4, 937-978. MR 3519989, https://doi.org/10.1007/s00209-016-1628-7
  • [OS] D. Orr and M. Shimozono, Specialization of nonsymmetric Macdonald-Koornwinder polynomials, to appear in J. Algebraic Combin., DOI 10.1007/s10801-017-0770-6. 11
  • [Pa] Paolo Papi, A characterization of a special ordering in a root system, Proc. Amer. Math. Soc. 120 (1994), no. 3, 661-665. MR 1169886, https://doi.org/10.2307/2160454
  • [Pe] D. Peterson, Quantum cohomology of $ G/P$, lecture notes, M.I.T., Spring 1997.
  • [Po] Alexander Postnikov, Quantum Bruhat graph and Schubert polynomials, Proc. Amer. Math. Soc. 133 (2005), no. 3, 699-709. MR 2113918, https://doi.org/10.1090/S0002-9939-04-07614-2
  • [RY] Arun Ram and Martha Yip, A combinatorial formula for Macdonald polynomials, Adv. Math. 226 (2011), no. 1, 309-331. MR 2735761, https://doi.org/10.1016/j.aim.2010.06.022

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B37, 33D52, 81R60, 81R50

Retrieve articles in all journals with MSC (2010): 17B37, 33D52, 81R60, 81R50


Additional Information

Satoshi Naito
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan
Email: naito@math.titech.ac.jp

Fumihiko Nomoto
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8551, Japan
Email: nomoto.f.aa@m.titech.ac.jp

Daisuke Sagaki
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Email: sagaki@math.tsukuba.ac.jp

DOI: https://doi.org/10.1090/tran/7114
Received by editor(s): May 8, 2016
Received by editor(s) in revised form: October 21, 2016
Published electronically: November 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society