Algebras and Representation Theory

, Volume 9, Issue 2, pp 161–199 | Cite as

Geometric and Combinatorial Realizations of Crystal Graphs

  • Alistair SavageEmail author


For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A n (1) , we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids.


affine Lie algebras Lusztig's quiver variety Nakajima's quiver variety crystal graphs Young tableaux 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Path space realization of the basic representation of An(1), In: V. G. Kac (ed.), Infinite Dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys. 7, World Scientific, Singapore, 1989, pp. 108–123. Google Scholar
  2. 2.
    Date, E., Jimbo, M., Kuniba, A., Miwa, T. and Okado, M.: Paths, Maya diagrams and representations of \(\widehat{\mathfrak{sl}}(r,\mathbb{C})\) , Adv. Stud. Pure Math. 19 (1989), 149–191. MathSciNetGoogle Scholar
  3. 3.
    Frenkel, I. B. and Savage, A.: Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics, Internat. Math. Res. Notices 28 (2003), 1521–1548. MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hong, J. and Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases, Amer. Math. Soc., Providence, 2000. Google Scholar
  5. 5.
    Kang, S.-J.: Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. London Math. Soc. 86 (2003), 26–69. CrossRefGoogle Scholar
  6. 6.
    Kashiwara, M. and Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 (1994), 295–345. MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kashiwara, M. and Saito, Y.: Geometric construction of crystal bases, Duke Math. J. 89(1) (1997), 9–36. MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping, J. Amer. Math. Soc. (4) (1991), 365–421. Google Scholar
  9. 9.
    Nakajima, H.: Homology of moduli spaces of instantons on ALE spaces, J. Differential Geom. 40 (1994), 105–127. zbMATHMathSciNetGoogle Scholar
  10. 10.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J. 76(2) (1994), 365–416. zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Nakajima, H.: Quiver varieties and Kac–Moody algebras, Duke Math. J. 91(3) (1998), 515–560. zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type An, Dn, Contemp. Math. 325 (2003), 141–160. zbMATHMathSciNetGoogle Scholar
  13. 13.
    Saito, Y.: Crystal bases and quiver varieties, Math. Ann. 324(4) (2002), 675–688. zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Savage, A.: A geometric realization of spin representations and Young diagrams from quiver varieties, arXiv:math.AG/0307018. Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Bahen Centre for Information TechnologyUniversity of TorontoTorontoCanada

Personalised recommendations