Integrality of two variable Kostka functions.

*(English)*Zbl 0876.05098In 1988, Ian Macdonald introduced a two-parameter family of symmetric polynomials, \(J_{\lambda}(x;q,t)\), which interpolate among many families of classical symmetric polynomials, see I. M. Macdonald [Symmetric functions and Hall polynomials, 2nd ed. (1995; Zbl 0824.05059)]. Among these are the Hall-Littlewood polynomials, \(Q_{\lambda}(x;t) = J_{\lambda}(x;0,t)\), and the Jack polynomials, \(\lim_{t\to 1} J_{\lambda}(x;t^{\alpha},t)\). The Hall-Littlewood polynomials are orthogonal with respect to a scalar product, \(\langle , \rangle_t\), for which \(K_{\lambda \mu}(t) = \langle s_{\lambda},Q_{\lambda}(x;t)\rangle_t\) was shown by A. Lascoux and M.-P. Schützenberger [C. R. Acad. Sci., Paris, Sér. A 286, 323-324 (1978; Zbl 0374.20010)] to be a polynomial with non-negative integral coefficients. Macdonald conjectured that \(\langle s_{\lambda},J_{\lambda}(x;q,t)\rangle_t\) would also be a polynomial in \(q\) and \(t\) with non-negative integral coefficients. In this paper, the author proves that it is a polynomial in \(q\) and \(t\). The proof proceeds by analogy with the proof of a similar conjecture for Jack polynomials. Other proofs of this conjecture have also appeared including one by A. M. Garsia and G. Tesler [Adv. Math. 123, No. 2, 144-222, Art. No. 0071 (1996; Zbl 0865.05075)], another by S. Sahi [Int. Math. Res. Not. 1996, No. 10, 457-471 (1996; Zbl 0861.05063)], and a third by Kirillov and Noumi [Affine Hecke algebras and raising operators for Macdonald polynomials, preprint].

Reviewer: D.M.Bressoud (St.Paul)