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A combinatorial model for crystals of Kac-Moody algebras

Authors: Cristian Lenart and Alexander Postnikov
Journal: Trans. Amer. Math. Soc. 360 (2008), 4349-4381
MSC (2000): Primary 17B67; Secondary 22E46, 20G42
Published electronically: February 27, 2008
MathSciNet review: 2395176
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Abstract: We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model. We describe crystal graphs and give a Littlewood-Richardson rule for decomposing tensor products of irreducible representations. The new model is based on the notion of a $ \lambda$-chain, which is a chain of positive roots defined by certain interlacing conditions.

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  • [Ch] C. Chevalley.
    Sur les dcompositions cellulaires des espaces $ G/B$.
    In Algebraic Groups and Generalizations: Classical Methods, volume 56 Part 1 of Proceedings and Symposia in Pure Mathematics, pages 1-23. Amer. Math. Soc., Providence, RI, 1994. MR 1278698 (95e:14041)
  • [De] V. V. Deodhar.
    Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function.
    Invent. Math., 39:187-198, 1977. MR 0435249 (55:8209)
  • [Dyer] M. J. Dyer.
    Hecke algebras and shellings of Bruhat intervals.
    Compositio Math., 89(1):91-115, 1993. MR 1248893 (95c:20053)
  • [Fu] W. Fulton.
    Young Tableaux, volume 35 of London Math. Soc. Student Texts.
    Cambridge Univ. Press, Cambridge and New York, 1997. MR 1464693 (99f:05119)
  • [GL] S. Gaussent and P. Littelmann.
    LS-galleries, the path model and MV-cycles.
    Duke Math. J.,127:35-88, 2005. MR 2126496 (2006c:20092)
  • [Hu] J. E. Humphreys.
    Reflection Groups and Coxeter Groups, volume 29 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [Jos] A. Joseph.
    Quantum Groups and Their Primitive Ideals,
    Springer-Verlag, New York, 1994. MR 1315966 (96d:17015)
  • [Kac] V. G. Kac.
    Infinite Dimensional Lie Algebras.
    Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
  • [Ka1] M. Kashiwara.
    Crystalizing the $ q$-analogue of universal enveloping algebras.
    Commun. Math. Phys., 133:249-260, 1990. MR 1090425 (92b:17018)
  • [Ka2] M. Kashiwara.
    On crystal bases of the $ q$-analogue of universal enveloping algebras.
    Duke Math. J., 63:465-516, 1991. MR 1115118 (93b:17045)
  • [Ka3] M. Kashiwara.
    Crystal bases of modified quantized enveloping algebra.
    Duke Math. J., 73:383-413, 1994. MR 1262212 (95c:17024)
  • [Kos] B. Kostant.
    Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra.
    Invent. Math., 158:181-226, 2004. MR 2090363 (2005m:17007)
  • [Ku] S. Kumar.
    Kac-Moody Groups, Their Flag Varieties and Representation Theory, volume 204 of Progress in Mathematics.
    Birkhäuser Boston Inc., Boston, MA, 2002. MR 1923198 (2003k:22022)
  • [La] V. Lakshmibai.
    Bases for quantum Demazure modules.
    In Representations of Groups (Banff, AB, 1994), volume 16 of CMS Conf. Proc., pages 199-216. Amer. Math. Soc., Providence, RI, 1995. MR 1357200 (96i:17014)
  • [LS] V. Lakshmibai and C. S. Seshadri.
    Standard monomial theory.
    In Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pages 279-322, Madras, 1991. Manoj Prakashan. MR 1131317 (92k:14053)
  • [LSc] A. Lascoux and M.-P. Schtzenberger.
    Keys and standard bases.
    In D. Stanton, editor, Invariant Theory and Tableaux, volume 19 of The IMA Vol. in Math. and Its Appl., pages 125-144, Berlin-Heidelberg-New York, 1990. Springer-Verlag. MR 1035493 (91c:05198)
  • [Le] C. Lenart.
    On the combinatorics of crystal graphs, I. Lusztig's involution.
    Adv. Math. 211:204-243, 2007. MR 2313533
  • [LP] C. Lenart and A. Postnikov.
    Affine Weyl groups in $ K$-theory and representation theory.
    Int. Math. Res. Not. 2007, no. 12, Art. ID rnm038, 65pp. MR 2344548
  • [Li1] P. Littelmann.
    A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras.
    Invent. Math., 116:329-346, 1994. MR 1253196 (95f:17023)
  • [Li2] P. Littelmann.
    Paths and root operators in representation theory.
    Ann. of Math. (2), 142:499-525, 1995. MR 1356780 (96m:17011)
  • [Li3] P. Littelmann.
    Characters of representations and paths in $ \mathfrak{h}^*_\mathbb{R}.$
    Proc. Sympos. Pure Math., 61:29-49, 1997. MR 1476490 (98j:17024)
  • [Lu] G. Lusztig.
    Canonical bases arising from quantized enveloping algebras. II.
    Progr. Theoret. Phys. Suppl., (102):175-201, 1991. MR 1182165 (93g:17019)
  • [PR] H. Pittie and A. Ram.
    A Pieri-Chevalley formula in the $ K$-theory of a $ G/B$-bundle.
    Electron. Res. Announc. Amer. Math. Soc., 5:102-107, 1999. MR 1701888 (2000d:14052)
  • [Shi] J.-Y. Shi.
    Sign type corresponding to an affine Weyl group.
    J. London Math. Soc. (2) 35:56-74, 1987. MR 871765 (88g:20103b)
  • [Ste] J. R. Stembridge.
    Combinatorial models for Weyl characters.
    Adv. Math., 168:96-131, 2002. MR 1907320 (2003j:17007)

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Additional Information

Cristian Lenart
Affiliation: Department of Mathematics and Statistics, State University of New York at Albany, Albany, New York 12222

Alexander Postnikov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): November 28, 2005
Received by editor(s) in revised form: September 1, 2006
Published electronically: February 27, 2008
Additional Notes: The first author was supported by National Science Foundation grant DMS-0403029 and by SUNY Albany Faculty Research Award 1039703
The second author was supported by National Science Foundation grant DMS-0201494 and by an Alfred P. Sloan Foundation research fellowship
Article copyright: © Copyright 2008 American Mathematical Society

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