## Integrality of two variable Kostka functions.(English)Zbl 0876.05098

In 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].

### MSC:

 05E05 Symmetric functions and generalizations 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 20C30 Representations of finite symmetric groups

### Citations:

Zbl 0824.05059; Zbl 0374.20010; Zbl 0865.05075; Zbl 0861.05063
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