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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Crystal bases and combinatorics of infinite rank quantum groups
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by Cédric Lecouvey PDF
Trans. Amer. Math. Soc. 361 (2009), 297-329 Request permission

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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 297-329
  • MSC (2000): Primary 05-02, 17B10, 17B37, 17B67
  • DOI: https://doi.org/10.1090/S0002-9947-08-04480-2
  • MathSciNet review: 2439408