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Various Representations of the Generalized Kostka Polynomials

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The Andrews Festschrift

Abstract

The generalized Kostka polynomials KλR(q) are labeled by a partition λ and a sequence of rectangles R. They are q-analogues of multiplicities of the finite-dimensional irreducible representation W(λ) of gIn with highest weight λ in the tensor product W (R 1) ⊗ … ⊗ W (RL). We review several representations of the generalized Kostka polynomials, such as the charge, path space, quasi-particle and bosonic representation. In addition we describe a bijection between Littlewood—Richardson tableaux and rigged configurations, and sketch a proof that it preserves the appropriate statistics. This proves in particular the equality of the quasi-particle and charge representation of the generalized Kostka polynomials.

Supported by the “Stichting Fundamenteel Onderzoek der Materie”.

Partially supported by NSF grant DMS-9800941.

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Author information

Author notes

  1. Anne Schilling

    Present address: Department of Mathematics, M.I.T., Cambridge, MA, 02139, USA

Authors and Affiliations

  1. Division of Mathematics, Graduate School of Science, Hokkaido University, Sapporo, 060-0810, Japan

    Anatol N. Kirillov

  2. Instituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE, Amsterdam, The Netherlands

    Anne Schilling

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061-0123, USA

    Mark Shimozono

Authors
  1. Anatol N. Kirillov
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  2. Anne Schilling
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  3. Mark Shimozono
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Editor information

Editors and Affiliations

  1. Département de mathématique, Université Louis Pasteur, 7, rue René-Descartes, 67084, Strasbourg, France

    Dominique Foata

  2. I.R.M.A. et C.N.R.S., Université Louis Pasteur, 7, rue René-Descartes, 67084, Strasbourg, France

    Guo-Niu Han

Additional information

Dedicated to George Andrews on the occasion of his sixtieth birthday.

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© 2001 Springer-Verlag Berlin Heidelberg

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Kirillov, A.N., Schilling, A., Shimozono, M. (2001). Various Representations of the Generalized Kostka Polynomials. In: Foata, D., Han, GN. (eds) The Andrews Festschrift. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56513-7_10

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  • DOI: https://doi.org/10.1007/978-3-642-56513-7_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41491-9

  • Online ISBN: 978-3-642-56513-7

  • eBook Packages: Springer Book Archive

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