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The Selberg-Jack symmetric functions. (English) Zbl 0885.33009
The author proves a generalization of Selberg’s integral evaluation [Norsk Mat. Tidsskr. 26, 71-79 (1944; Zbl 0063.06870)] in which the integrand is multiplied by renormalized Jack polynomials. This encompasses several earlier generalizations of Selberg’s integral due to K. Aomoto [SIAM J. Math. Anal. 18, 545-549 (1987; Zbl 0639.33001)], K. W. J. Kadell [SIAM J. Math. Anal. 19, No. 4, 944-968 (1988; Zbl 0643.33003), and Trans. Am. Math. Soc. 310, No. 2, 535-553 (1988; Zbl 0706.33015)], and D. St. P. Richards [Analogs and extensions of Selberg’s integral, in \(q\)-series and partitions, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 18, 109-137 (1989; Zbl 0697.33007)]. Kadell’s identity can be used to define the Jack polynomials, and the author exploits this to prove Stanley’s extension of the Pieri formula, Macdonald’s extension of the duality property, and the combinatorial representation of Jack polynomials. The author also obtains generalizations of the constant term identities of Morris for \(A_n\) and Macdonald-Morris for \(BC_n\).

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