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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 $(\alpha=1)$ and zonal polynomials $(\alpha=2)$. This paper, in the spirit of I. G. Macdonald’s book {\it 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
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##### References:
 [1] Constantine, A. G.: Some noncentral distribution problems in multivariate analysis. Ann. math. Statist. 34, 1270-1285 (1963) · Zbl 0123.36801 [2] P. Diaconis and E. Lander, Some formulas for zonal polynomials, in preparation. [3] Foulkes, H. O.: A survey of some combinatorial aspects of symmetric functions. Permutations (1974) · Zbl 0282.05004 [4] Hanlon, P.: Jack symmetric functions and some combinatorial properties of Young symmetrizers. J. combin. Theory ser. A 47, 37-70 (1988) · Zbl 0641.20010 [5] Jack, H.: A class of symmetric polynomials with a parameter. Proc. roy. Soc. Edinburgh sect. A 69, 1-17 (1969-1970) [6] James, A. T.: Zonal polynomials of the real positive definite symmetric matrices. Ann. of math. 74, 456-469 (1961) · Zbl 0104.02803 [7] James, A. T.: Distributions of matrix variables and latent roots derived from normal samples. Ann. math. Statist. 35, 475-501 (1964) · Zbl 0121.36605 [8] James, A. T.: Calculation of zonal polynomial coefficients by use of the Laplace-Beltrami operator. Ann. math. Statist. 39, 1711-1718 (1968) · Zbl 0177.47406 [9] H. B. Kushner, The linearization of the product of two zonal polynomials, preprint. · Zbl 0642.33024 [10] Kerber, A.; Thürlings, K. -J: Symmetrieklassen von funktionen und ihre abzählungstheorie. Bayreuther mathematische schriften, No. Heft 15 (1983) [11] Littlewood, D. E.: The theory of group characters. (1950) · Zbl 0038.16504 [12] Macdonald, I. G.: Symmetric functions and Hall polynomials. (1979) · Zbl 0487.20007 [13] Macdonald, I. G.: Commuting differential equations and zonal spherical functions. Lecture notes in math. 1271, 189-200 (1987) [14] [M3]I. G. Macdonald, [M1], 2nd ed., to appear. [15] Specht, W.: Die characktere der symmetrichen gruppe. Math. zeit. 73, 312-329 (1960) · Zbl 0096.01902 [16] Takemura, A.: Zonal polynomials. Institute of mathematical statistics, lecture notes-monograph series 4 (1984)