Generalized Young Walls for Classical Lie Algebras

Abstract

In this paper, we introduce an new combinatorial model, which we call generalized Young walls for classical Lie algebras, and we give two realizations of the crystal B() over classical Lie algebras using generalized Young walls. Also, we construct natural crystal isomorphisms between generalized Young wall realizations and other realizations, for example, monomial realization, polyhedral realization and tableau realization. Moreover, as applications, we obtain a crystal isomorphism between two different polyhedral realizations of B().

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References

  1. 1.

    Cliff, G.: Crystal Bases and Young Tableaux. J. Algebra 202, 10–35 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases Graduate Studies in Mathematics, vol. 42. American Mathematical Society, Providence (2002)

    Google Scholar 

  3. 3.

    Hong, J., Lee, H.: Young tableaux and crystal B() for finite simple Lie algebras. J Algebra 320, 3680–3693 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Jeong, K., Kang, S.-J., Kim, J.-A., Shin, D.-U.: Crystals and Nakajima monomials for quantum generalized Kac-Moody algebras. J. Algebra 319, 3732–3751 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Jimbo, M., Misra, K.C., Miwa, T., Okado, M.: Combinatorics of representations of \({U}_{q}(\widehat {\mathfrak {sl}}(n))\) at q = 0. Comm. Math. Phys. 136, 543–566 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls. Proc. Lond. Math. Soc. 86, 29–69 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of Verma modules for the quantum affine Lie algebras. Composito Math. 92, 299–325 (1994)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Crystal bases of Verma modules for the quantum affine Lie algebras. Int. J. Mod. Phys. A. Suppl. 1A, 449–484 (1992)

    Article  MATH  Google Scholar 

  9. 9.

    Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke Math. J. 68, 499–607 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Kang, S.-J., Kim, J.-A., Shin, D.-U.: Monomial realization of crystal bases for special linear Lie algebras. J. Algebra 274, 629–642 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Kang, S.-J., Kim, J.-A., Shin, D.-U.: Crystal bases for quantum classical algebras and Nakajima’s monomials. Publ. Res. Inst. Math. Sci. 40, 758–791 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Kang, S.-J., Kim, J.-A., Shin, D.-U: Modified Nakajima monomials and the crystal B(). J Algebra 308, 524–535 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kang, S.-J., Kim, J.-A., Lee, H., Shin, D.-U.: Young wall realization of crystal bases for classical Lie algebras. Trans. Amer. Math. Soc. 356, 2349–2378 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kang, S.-J., Lee, H.: Crystal bases for quantum affine algebras and Young walls. J. Algebra 322, 1979–1999 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kang, S.-J., Lee, K.-H., Ryu, H., Salisbury, B.: A combinatorial description of the affine Gindikin-Karpelevich formula of type \(A_{n}^{(1)}\), preprint, arXiv:1203.1640, to appear in Proceedings of the Symposium Pure Mathematics

  16. 16.

    Kang, S.-J., Misra, K.C.: Crystal bases and tensor product decompositions of U q(G 2)-modules. J. Algebra 163, 675–691 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Kashiwara, M.: Crystalizing the q-analogue of universal enveloping algebras. Comm. Math. Phys. 133, 249–260 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Kashiwara, M.: The crystal base and Littlemanns refined Demazure character formula. Duke Math. J. 71, 839–858 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Kashiwara, M.: Realizations of crystals. Amer. Math. Soc. 325, 133–139 (2003)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, 295–345 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Kim, J.-A.: Monomial realization of crystal graphs for \(U_{q}(A_{n}^{(1)})\). Math. Ann. 332, 17–35 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kim, J.-A., Shin, D.-U.: Correspondence between Young walls and Young tableaux and its application. J. Algebra 282, 728–757 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kim, J.-A., Shin, D.-U.: Monomial realization of crystal bases B() for the quantum finite algebras. Algebr. Represent. Theory 11, 93–105 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Kim, J.-A., Shin, D.-U.: Generalized Young walls and crystal bases for quantum affine algebra of type A. Proc. Amer. Math Soc. 138, 3877–3889 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Kim, J.-A., Shin, D.-U.: Zigzag strip bundles and crystals. J. Combin. Theory Ser. A 120, 1087–1115 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Kim, J.-A., Shin, D.-U.: Zigzag strip bundles and the crystal B() for quantum affine algebras. Comm. Algebra 43, 1983–2004 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Kim, J.-A., Shin, D.-U.: Zigzag strip bundles and highest weight crystals. J. Algebra 412, 15–50 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Kim, J.-A., Shin, D.-U.: Zigzag strip bundle realization of B0) over \(U_{q}(C_{n}^{(1)})\). Algebr. Represent. Theory 19, 1423–1436 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Lee, K.-H., Salisbury, B.: A combinatorial description of the Gindikin-Karpelevich formula in type A. J. Combin. Theory Ser. A 119, 1081–1094 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Lee, K.-H., Salisbury, B.: Young tableaux, canonical bases and the Gindikin-Karpelevich formula. J Korean Math. Soc. 51, 289–309 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Littelmann, P.: Crystal graphs and Young tableaux. J. Algebra 175, 65–87 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Littelmann, P.: Paths and root operators in representation theory. Ann. Math. 142, 499–525 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Littelmann, P.: A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math. 116, 329–346 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type A n, D n, Contemporary Mathematics, vol. 325, pp. 141–160. American Mathematical Society, Providence (2003)

    Google Scholar 

  36. 36.

    Nakashima, T.: Polyhedral realizations of crystal bases for integrable highest weight modules. J. Algebra 219, 571–597 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Nakashima, T., Zelevinsky, A.: Polyhedral realization of crystal bases for quantized Kac-Moody algebras. Adv. Math. 131, 253–278 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Shin, D.-U.: Crystal Bases and Monomials for U q(G 2)-modules. Comm. Algebra 34, 129–142 (2006)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Shin, D.-U.: Polyhedral realization of crystal bases for generalized Kac-Moody algebras. J. Lond. Math. Soc. 77, 273–286 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Shin, D.-U.: Polyhedral realization of the highest weight crystals for generalized Kac-Moody algebras. Trans. Amer. Math. Soc. 360, 6371–6387 (2008)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The first author’s research was supported by NRF Grant # 2015R1C1A2A01053319.

The second author’s research was supported by NRF Grant #2017R1D1A1B03028399.

This work was initiated while the authors were visiting the Department of Mathematics, University of Connecticut, from August 2013 to July 2014, and part of this work was done during they visited Korea Institute for Advanced Study, in the fall of 2016. They would like to express their sincere gratitude to the staff of the Department of Mathematics, University of Connecticut and Korea Institute for Advanced Study for their hospitality and support.

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Correspondence to Dong-Uy Shin.

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Presented by Anne Schilling.

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Kim, J., Shin, D. Generalized Young Walls for Classical Lie Algebras. Algebr Represent Theor 22, 345–373 (2019). https://doi.org/10.1007/s10468-018-9770-z

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Keywords

  • Crystals
  • Generalized Young walls
  • Tableaux
  • Nakajima monomials
  • Kashiwara embeddings

Mathematics Subject Classification (2010)

  • Primary 81R50
  • Secondary 17B37