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Polynomiality of the q, t-Kostka revisited

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Abstract

The polynomiality problem for the q, t-Kostka coefficients [11] was posed by Macdonald in the fall 1988 meeting of the Lotharingian seminar. It remained open for quite a few years, when suddenly, in 1996, several proofs of varied difficulty appeared in a period of only a few months. At the present there are three basically different approaches to proving the polynomiality of the q, t-Kostka coefficients:

  1. (1)

    via plethystic formulas (Garsia-Tesler [4], Garsia-Remmel [3]);

  2. (2)

    via vanishing properties (Sahi [13], Knop [7, 8]);

  3. (3)

    via Rodrigues formulas (Lapointe-Vinet [10], Kirillov-Noumi [6]).

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References

  1. Garsia, A.M., Haiman, M. (1996): Some natural bigraded S n-modules and q, t-Kostka coefficients. The Foata Festschrift. Electron. J. Combin. 3,no. 2, Research Paper 24, approx. 60 pp. (electronic)

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Garsia, A.M., Zabrocki, M. (2001). Polynomiality of the q, t-Kostka revisited. In: Crapo, H., Senato, D. (eds) Algebraic Combinatorics and Computer Science. Springer, Milano. https://doi.org/10.1007/978-88-470-2107-5_20

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  • DOI: https://doi.org/10.1007/978-88-470-2107-5_20

  • Publisher Name: Springer, Milano

  • Print ISBN: 978-88-470-2159-4

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