Jack symmetric functions and some combinatorial properties of Young symmetrizers (Q1100552)

The Jack symmetric functions, \(J_{\lambda}(\alpha;x)\) are symmetric functions indexed by partitions \(\lambda\) which depend on an indeterminate \(\alpha\) [see \textit{H. Jack}, Proc. R. Soc. Edinb., Sect. A 69, 1--18 (1970; Zbl 0198.04606)]. These functions are of recent interest because they are common generalizations of two sets of spherical functions, the Schur functions \(s_{\lambda}(x)\) and the zonal polynomials \(Z_{\lambda}(x)\) [see \textit{A. T. James}, Ann. Math., II. Ser. 74, 456--469 (1961; Zbl 0104.02803)]. Write \(J_{\lambda}(\alpha,x)\) in terms of the monomial symmetric functions \(J_{\lambda}(\alpha,x)=\sum_{\mu}C_{\lambda \mu}(\alpha)m_{\mu}(x)\). This paper stems from an effort to settle the still open question of whether the coefficients \(C_{\lambda \mu}(\alpha)\) are polynomials in \(\alpha\).