Integrality of Two Variable Kostka Functions
نویسنده
چکیده
Macdonald defined in [M1] a remarkable class of symmetric polynomials Pλ(x; q, t) which depend on two parameters and interpolate between many families of classical symmetric polynomials. For example Pλ(x; t) = Pλ(x; 0, t) are the Hall-Littlewood polynomials which themselves specialize for t = 0 to Schur functions sλ. Also Jack polynomials arise by taking q = t and letting t tend to 1. The Hall-Littlewood polynomials are orthogonal with respect to a certain scalar product 〈·, ·〉t. The scalar products Kλμ = 〈sλ, sμ〉0 are known as Kostka numbers. Since it has an interpretation as a weight multiplicity of a GLn-representation, it is a natural number. Also the scalar products Kλμ(t) = 〈Pλ(x; t), sμ〉t are important polynomials in t. Their coefficients have been proven to be natural numbers by Lascoux-Schützenberger [LS]. Macdonald conjectured in [M1] that even the scalar products
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