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SOME CLASSICAL EXPANSIONS FOR KNOP-SAHI AND MACDONALD POLYNOMIALS

Jennifer Morse

unpublished

In recent simultaneous work, Knop and Sahi introduced a non-homogeneous non-symmetric polynomial G α (x; q, t) whose highest homogeneous component gives the non-symmetric Macdonald polynomial E α (x; q, t). Macdonald shows that for any composition α that rearranges to a partition λ, an appropriate Hecke algebra symmetrization of E α yields the Macdonald polynomial P λ (x; q, t). In the original papers all these polyno-mials are only shown to exist. No explicit expressions are given relating
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... given relating them to the more classical bases. Our basic discovery here is that G α (x; q, t) appears to have surprisingly elegant expansions in terms of the polynomials Z α (x 1 ,. .. , x n ; q) = n i=1 (x i ; q) αi. In this paper we present the first results obtained in the problem of determining the connection coefficients relating these bases. In particular we give a solution to the problem of two variables. Our proofs rely on the theory of basic hypergeometric series and reveal a deep connection between this classical subject and the theory of Macdonald polynomials. Introduction The Macdonald basis {P λ (x; q, t)} λ has recently become an intensive subject of study as a result of the many difficult conjectures that surround it. Its importance in the development of symmetric function theory is now widely recognized. In addition to specializing to several fundamental bases, (such as the Schur, the Hall-Littlewood, the Zonal, the Jack) its has been conjectured to occur in a natural way [1] in representation theory and in some problems of particle mechanics [8]. One of the difficulties encountered in its study is the absence of explicit formulas expressing P λ (x; q, t) in terms of more familiar bases. In fact, the connection coefficients relating a rescaled version of {P λ (x; q, t)} λ to the modified Schur basis {S λ [X(1 − t); q, t]} λ have only recently been shown to be polynomial functions of q, t ([2],[4],[5],[7]). Macdonald in [12] shows that each P λ (x; q, t) decomposes into a sum of homogeneous non-symmetric polynomials E α (x; q, t) indexed by compositions. More precisely, if α is any composition that rearranges to λ then

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