A Recursion and a Combinatorial Formula for Jack Polynomials

The Jack polynomials Jλ(x; α) form a remarkable class of symmetric polynomials. They are indexed by a partition λ and depend on a parameter α. One of their properties is that several classical families of symmetric functions can be obtained by specializing α, e.g., the monomial symmetric functions mλ (α = ∞), the elementary functions eλ′ (α = 0), the Schur functions sλ (α = 1) and finally the two classes of zonal polynomials (α = 2, α = 1/2). The Jack polynomials can be defined in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical filtration of the ring symmetric functions or b) as simultaneous eigenfunctions of certain differential operators (the Sekiguchi-Debiard operators). Recently Opdam, [O], constructed a similar family Fλ(x; α) of non-symmetric polynomials. The index runs now through all compositions λ ∈ N. They are defined in a completely similar fashion, e.g., the Sekiguchi-Debiard operators are being replaced by the Cherednik differential-reflection operators (see section 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the Fλ together with two formulas to obtain Jλ from them. d) combinatorial formulas of both Jλ and Fλ in terms of certain generalized tableaux. There are many advantages of these new characterizations over the ones mentioned above. In a) and b), the existence of functions with these properties is not obvious and requires a proof whereas c) and d) could immediately serve as a definition of Jack polynomials.