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Generalized Weyl modules for twisted current algebras

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Abstract

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case.

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References

  1. V. Chari, B. Ion, and D. Kus, “Weyl modules for the hyperspecial current algebra,” Internat. Math. Res. Notices, 2015, 6470–6515 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. Chari, G. Fourier, and P. Senesi, “Weyl modules for the twisted loop algebras,” J. Algebra, 319, 5016–5038 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. Fourier and D. Kus, “Demazure and Weyl modules: The twisted current case,” Trans. Amer. Math. Soc., 365, 6037–6064 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Chari and A. Pressley, “Weyl modules for classical and quantum affine algebras,” Represent. Theory, 5, 191–223 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Chari and S. Loktev, “Weyl, Demazure, and fusion modules for the current algebra of sl r+1,” Adv. Math., 207, 928–960 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Fourier and P. Littelmann, “Tensor product structure of affine Demazure modules and limit constructions,” Nagoya Math. J., 182, 171–198 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Fourier and P. Littelmann, “Weyl modules, Demazure modules, KR-modules, crystals, fusion products, and limit constructions,” Adv. Math., 211, 566–593 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Sanderson, “On the connection between Macdonald polynomials and Demazure characters,” J. Algebraic Combin., 11, 269–275 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Ion, “Nonsymmetric Macdonald polynomials and Demazure characters,” Duke Math. J., 116, 299–318 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Chari and B. Ion, “BGG reciprocity for current algebras,” Compos. Math., 151, 1265–1287 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1995).

    MATH  Google Scholar 

  12. I. G. Macdonald, “Affine Hecke algebras and orthogonal polynomials,” Astérisque, 237, 189–207 (1996).

    MathSciNet  MATH  Google Scholar 

  13. K. Naoi, “Weyl modules, Demazure modules, and finite crystals for non-simply laced type,” Adv. Math., 229, 875–934 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Cherednik, “Nonsymmetric Macdonald polynomials,” Internat. Math. Res. Notices, 1995, 483–515 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Cherednik, Double Affine Hecke Algebras (London Math. Soc. Lect. Note Ser., Vol. 319), Cambridge Univ. Press, Cambridge (2006).

    MATH  Google Scholar 

  16. D. Orr and M. Shimozono, “Specializations of nonsymmetric Macdonald–Koornwinder polynomials,” arXiv: 1310.0279v2 [math.QA] (2013).

    Google Scholar 

  17. E. Feigin and I. Makedonskyi, “Generalized Weyl modules, alcove paths, and Macdonald polynomials,” arXiv: 1512.03254v6 [math.RT] (2015).

    MATH  Google Scholar 

  18. E. Feigin, I. Makedonskyi, and D. Orr, “Generalized Weyl modules and nonsymmetric q-Whittaker functions,” arXiv:1605.01560v1 [math.RT] (2016).

    Google Scholar 

  19. I. Cherednik and D. Orr, “Nonsymmetric difference Whittaker functions,” Math. Z., 279, 879–938 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Cherednik and D. Orr, “One-dimensional nil-DAHA and Whittaker functions: II,” Transform. Groups, 18, 23–59 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Ram and M. Yip, “A combinatorial formula for Macdonald polynomials,” Adv. Math., 226, 309–331 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Gaussent and P. Littelmann, “LS galleries, the path model, and MV cycles,” Duke Math. J., 127, 35–88 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Naito and D. Sagaki, “Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials,” Math. Z., 283, 937–978 (2016); arXiv:1404.2436v3 [math.QA] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Naito, F. Nomoto, and D. Sagaki, “Specialization of nonsymmetric Macdonald polynomials at t = ∞ and Demazure submodules of level-zero extremal weight modules,” arXiv:1511.07005v2 [math.QA] (2015).

    Google Scholar 

  25. G. Lusztig, “Hecke algebras and Jantzen’s generic decomposition patterns,” Adv. Math., 37, 121–164 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  26. F. Brenti, S. Fomin, and A. Postnikov, “Mixed Bruhat operators and Yang–Baxter equations for Weyl groups,” Internat. Math. Res. Notices, 1999, 419–441 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Feigin and I. Makedonskyi, “Weyl modules for osp(1, 2) and nonsymmetric Macdonald polynomials,” arXiv: 1507.01362v1 [math.RT] (2015).

    MATH  Google Scholar 

  28. C. Lenart, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, “A uniform model for Kirillov–Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph,” Internat. Math. Res. Notices, 2015, 1848–1901 (2015).

    MathSciNet  MATH  Google Scholar 

  29. C. Lenart and A. Lubovsky, “A uniform realization of the combinatorial R-matrix,” arXiv:1503.01765v1 [math.RT] (2015).

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    I. A. Makedonskyi & E. B. Feigin

  2. Max Planck Institute for Mathematics, Bonn, Germany

    I. A. Makedonskyi

  3. Skolkovo Institute of Science and Technology, Moscow, Russia

    E. B. Feigin

Authors
  1. I. A. Makedonskyi
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  2. E. B. Feigin
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Corresponding author

Correspondence to I. A. Makedonskyi.

Additional information

This research was supported by the Russian Academic Excellence Project 5-100, and the research reported in Secs. 1, 2, and 3 was supported by the RSF-DFG (Grant No. 16-41-01013).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 192, No. 2, pp. 284–306, August, 2017.

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Makedonskyi, I.A., Feigin, E.B. Generalized Weyl modules for twisted current algebras. Theor Math Phys 192, 1184–1204 (2017). https://doi.org/10.1134/S0040577917080086

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