Table of Contents
ISRN Combinatorics
Volume 2013 (2013), Article ID 431024, 17 pages
http://dx.doi.org/10.1155/2013/431024
Research Article

Crystal Bases as Tuples of Integer Sequences

Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany

Received 31 January 2013; Accepted 19 February 2013

Academic Editors: M. Caramia and R. Dondi

Copyright © 2013 Deniz Kus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Kashiwara, “On crystal bases of the Q-analogue of universal enveloping algebras,” Duke Mathematical Journal, vol. 63, no. 2, pp. 465–516, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” Journal of Algebra, vol. 165, no. 2, pp. 295–345, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  3. P. Littelmann, “A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras,” Inventiones Mathematicae, vol. 116, no. 1–3, pp. 329–346, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Kashiwara and Y. Saito, “Geometric construction of crystal bases,” Duke Mathematical Journal, vol. 89, no. 1, pp. 9–36, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Nakajima, “Quiver varieties and tensor products,” Inventiones Mathematicae, vol. 146, no. 2, pp. 399–449, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Kashiwara, “Realizations of crystals,” in Combinatorial and Geometric Representation Theory (Seoul, 2001), vol. 325 of Contemporary Mathematics, pp. 133–139, American Mathematical Society, Providence, RI, USA, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S.-J. Kang, J.-A. Kim, and D.-U. Shin, “Monomial realization of crystal bases for special linear Lie algebras,” Journal of Algebra, vol. 274, no. 2, pp. 629–642, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S.-J. Kang, J.-A. Kim, and D.-U. Shin, “Crystal bases for quantum classical algebras and Nakajima's monomials,” Kyoto University. Research Institute for Mathematical Sciences. Publications, vol. 40, no. 3, pp. 757–791, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Meng, “Compression of Nakajima monomials in type A and C,” Journal of Algebraic Combinatorics, vol. 35, no. 4, pp. 649–690, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. G. Fourier and D. Kus, “Compression of Nakajima monomials in type B and D,” In preparation.
  11. J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, vol. 42 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2002. View at MathSciNet
  12. J. R. Stembridge, “A local characterization of simply-laced crystals,” Transactions of the American Mathematical Society, vol. 355, no. 12, pp. 4807–4823, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. V. I. Danilov, A. V. Karzanov, and G. A. Koshevoy, “B2-crystals: axioms, structure, models,” Journal of Combinatorial Theory A, vol. 116, no. 2, pp. 265–289, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type An,Dn,” in Combinatorial and Geometric Representation Theory (Seoul, 2001), vol. 325 of Contemporary Mathematics, pp. 141–160, American Mathematical Society, Providence, RI, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J.-A. Kim, “Monomial realization of crystal graphs for Uq(An(1)),” Mathematische Annalen, vol. 332, no. 1, pp. 17–35, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. D. Kus, “Realization of affine type A Kirillov-Reshetikhin crystals via polytopes,” http://arxiv.org/abs/1209.6019.