Hook-lengths and Pairs of Compositions

The nonsymmetric Jack polynomials are defined to be the simultaneous eigenfunctions of a parametrized commuting set of first-order differentialdifference operators. These polynomials form a basis for the homogeneous polynomials and they are labeled by compositions, just like the monomials. The coefficients of the polynomials when expanded in the standard monomial basis are rational functions of the parameter. The poles are determined by certain hook-length products. Another way of locating the poles depends on possible degeneracies of the eigenvalues under the defining set of operators, when the parameter takes on certain negative rational values. This property can be described in an elementary geometric way. Here is an example: consider the two compositions (2,7,8,2,0,0) and (5,1,2,5,3,3), then the respective ranks (permutations of the index set {1,2,...,6} sorting the compositions) are (3,2,1,4,5,6) and (1,6,5,2,3,4), and the two vectors of differences (between the compositions and the ranks, respectively) are (-3,6,6,-3,-3,-3) and (2,-4,-4,2,2,2), which are parallel, with ratio -3/2. It is this parallelism property which is associated with the degeneracy of eigenvalues. For a given composition and associated negative rational number there is an algorithm for constructing another composition with the parallelism property and which is comparable to it in a certain partial order on compositions, derived from the dominance order. This paper presents the background on the polynomials and hook-lengths, and establishes the properties of the algorithm. There is a discussion of some open problems.