Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the combinatorics of crystal graphs, II. The crystal commutor


Author: Cristian Lenart
Journal: Proc. Amer. Math. Soc. 136 (2008), 825-837
MSC (2000): Primary 20G42; Secondary 17B10, 22E46
Published electronically: November 30, 2007
MathSciNet review: 2361854
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen.


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

  • 1. Arkady Berenstein and Andrei Zelevinsky, Canonical bases for the quantum group of type 𝐴ᵣ and piecewise-linear combinatorics, Duke Math. J. 82 (1996), no. 3, 473–502. MR 1387682, 10.1215/S0012-7094-96-08221-6
  • 2. Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, 10.1007/s002220000102
  • 3. N. Bourbaki.
    Groupes et Algèbres de Lie. Chp. IV-VI.
    Masson, Paris, 1981.
  • 4. V. G. Drinfel′d, Quasi-Hopf algebras, Algebra i Analiz 1 (1989), no. 6, 114–148 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 6, 1419–1457. MR 1047964
  • 5. William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
  • 6. André Henriques and Joel Kamnitzer, Crystals and coboundary categories, Duke Math. J. 132 (2006), no. 2, 191–216. MR 2219257, 10.1215/S0012-7094-06-13221-0
  • 7. Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966
  • 8. J. Kamnitzer.
    The crystal structure on the set of Mirković-Vilonen polytopes. Adv. Math., 215:66-93, 2007.
  • 9. J. Kamnitzer and P. Tingley.
    A definition of the crystal commutor using Kashiwara's involution.
    arXiv:math.QA/0610952.
  • 10. Masaki Kashiwara, Crystalizing the 𝑞-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425
  • 11. Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
  • 12. Masaki Kashiwara and Toshiki Nakashima, Crystal graphs for representations of the 𝑞-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345. MR 1273277, 10.1006/jabr.1994.1114
  • 13. Cristian Lenart, On the combinatorics of crystal graphs. I. Lusztig’s involution, Adv. Math. 211 (2007), no. 1, 204–243. MR 2313533, 10.1016/j.aim.2006.08.002
  • 14. C. Lenart and A. Postnikov.
    Affine Weyl groups in $ K$-theory and representation theory.
    Int. Math. Res. Not., Article ID rnm038, 65pp.
  • 15. C. Lenart and A. Postnikov.
    A combinatorial model for crystals of Kac-Moody algebras.
    arXiv:math.RT/0502147.
    To appear in Trans. Amer. Math. Soc.
  • 16. Peter Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499–525. MR 1356780, 10.2307/2118553
  • 17. Peter Littelmann, Characters of representations and paths in ℌ*_{𝐑}, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 29–49. MR 1476490, 10.1090/pspum/061/1476490
  • 18. G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175–201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, 10.1143/PTPS.102.175
  • 19. George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
  • 20. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282
  • 21. John R. Stembridge, Combinatorial models for Weyl characters, Adv. Math. 168 (2002), no. 1, 96–131. MR 1907320, 10.1006/aima.2001.2050
  • 22. M. A. A. van Leeuwen.
    An analogue of jeu de taquin for Littelmann's crystal paths.
    Sém. Lothar. Combin., 41:Art. B41b, 23 pp. (electronic), 1998.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20G42, 17B10, 22E46

Retrieve articles in all journals with MSC (2000): 20G42, 17B10, 22E46


Additional Information

Cristian Lenart
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222
Email: lenart@albany.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09244-1
Keywords: Crystals, root operators, coboundary category, commutor, Lusztig's involution, van Leeuwen's jeu de taquin.
Received by editor(s): February 16, 2007
Published electronically: November 30, 2007
Additional Notes: The author was supported by National Science Foundation grant DMS-0403029
Communicated by: Jim Haglund
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.