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Some combinatorial properties of Jack symmetric functions. (English) Zbl 0743.05072
Jack functions are symmetric functions parametrized by an indeterminate $\alpha$, which generalize Schur functions $\left(\alpha =1\right)$ and zonal polynomials $\left(\alpha =2\right)$. This paper, in the spirit of I. G. Macdonald’s book Symmetric functions and Hall polynomials [Clarendon Press, New York (1979; Zbl 0487.20007)], extends many of the basic results on Schur functions to Jack functions. Induced in the paper are results due to Macdonald which appear in the second edition of the above book. The results include a definition of a parametrized hook length, evaluations of the norm of a Jack function in terms of these hook lengths, and duality results relating conjugate shapes with the reciprocal of the parameter. Several conjectures are mentioned. Among these are a conjecture that the (normalized) Kostka number analogue is a polynomial with nonnegative integer coefficients and a conjecture that the Littlewood-Richardson coefficient analogue is a polynomial with nonnegative integer coefficients.

##### MSC:
 05E05 Symmetric functions and generalizations 20C30 Representations of finite symmetric groups 05A15 Exact enumeration problems, generating functions