A recursion and a combinatorial formula for Jack polynomials. (English) Zbl 0870.05076

Knop, Friedrich; Sahi, Siddhartha

Invent. Math. 128, No. 1, 9-22 (1997).

The paper deals with Jack polynomials \(J_\lambda(x;\alpha)\) and a similar family of (nonsymmetric) polynomials \(F_\lambda(x;\alpha)\), recently constructed by Opdam. Two characterizations are given: A recursion formula for \(F_\lambda\) and a combinatorial formula in terms of certain generalized tableaux for both \(F_\lambda\) and \(J_\lambda\). These characterizations are more explicit than the usual definitions as orthogonal family in the ring of symmetric functions, or as eigenfunctions of certain differential operators. Taking these characterizations as new definitions, existence of the polynomials is immediately clear; moreover, the conjecture of Macdonald (concerning the coefficients \(v_{\lambda\mu}(\alpha)\) of \(m_\mu\) in the expansion of \(J_\lambda(x;\alpha)\)) follows easily from the combinatorial formula.

Reviewer: M.Fulmek (Wien)

MSC:

05E05	Symmetric functions and generalizations
05E35	Orthogonal polynomials (combinatorics) (MSC2000)

Keywords:

Jack polynomials; symmetric functions; tableaux; Cherednik operators