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Crystal bases and combinatorics of infinite rank quantum groups

Author(s): Cédric Lecouvey
Journal: Trans. Amer. Math. Soc. 361 (2009), 297-329.
MSC (2000): Primary 05-02, 17B10, 17B37, 17B67
Posted: August 14, 2008
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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: 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
Posted: August 14, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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