Séminaire Lotharingien de Combinatoire, B32c (1994), 38 pp.

J. Désarménien, B. Leclerc and J.-Y. Thibon

Hall-Littlewood Functions and Kostka-Foulkes Polynomials in Representation Theory

Abstract. This paper presents a survey of recent applications of Hall-Littlewood functions and Kostka-Foulkes polynomials to the representation theory of the general linear group GL(n,C) and of the symmetric group S(n). The reviewed topics include the q-analogue of Kostant's partition function, vertex operators, generalized exponents of GL(n,C) and S(n)-harmonic polynomials. We also give a detailed description of the various combinatorial interpretations of Kostka-Foulkes polynomials. We conclude with the study of Hall-Littlewood functions at roots of unity, which provide a combinatorial description of certain plethysms.

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