Symmetric Jack Polynomials from Non–symmetric Theory

In (1.2) mκ(z) is the monomial symmetric function in the variables z1, . . . , zN , and the sum is over all partitions μ which have the same modulus as κ but are smaller in dominance ordering. The polynomials Pκ possess a host of special properties, and in fact form the natural basis for a class of symmetric multivariable orthogonal polynomials generalizing the classical orthogonal polynomials [7, 8, 2]. Although through the efforts of Macdonald [9], Stanley [12] and others, the theory of symmetric Jack polynomials is highly developed, many theorems seem difficult to prove. One reason for this is that the symmetric Jack polynomials are not the most fundamental polynomials in the theory – this title belongs to the non-symmetric Jack polynomials Eη := Eη(z;α), which were introduced [10] after the pioneering works of Macdonald and Stanley. For a given composition of non-negative integers η := (η1, . . . , ηN ), the polynomials Eη can be defined as the unique polynomial of the form Eη(z;α) = z η + ∑