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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Crystal bases and combinatorics of infinite rank quantum groups


Author: Cédric Lecouvey
Journal: Trans. Amer. Math. Soc. 361 (2009), 297-329
MSC (2000): Primary 05-02, 17B10, 17B37, 17B67
Published electronically: August 14, 2008
MathSciNet review: 2439408
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Abstract: The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independent of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits us in particular to obtain for each family of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bicrystal on these biwords. This RSK-correspondence yields also a plactic algebra and plactic Schur functions distinct for each infinite root system. Surprisingly, the algebras spanned by these plactic Schur functions are all isomorphic to the algebra of symmetric functions.


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Additional Information

Cédric Lecouvey
Affiliation: Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville, B.P. 699, 62228 Calais Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04480-2
PII: S 0002-9947(08)04480-2
Received by editor(s): May 12, 2006
Received by editor(s) in revised form: December 30, 2006, January 10, 2007, and January 17, 2007
Published electronically: August 14, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.