Hook-lengths and Pairs of Compositions

Charles F. Dunkl

Published 2004-10-21, updated 2005-05-20Version 2

The monomial basis for polynomials in N variables is labeled by compositions. To each composition there is associated a hook-length product, which is a product of linear functions of a parameter. The zeroes of this product are related to "critical pairs" of compositions; a concept defined in this paper. This property can be described in an elementary geometric way; for 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. For a given composition and zero of its hook-length product 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 motivation from the theory of nonsymmetric Jack polynomials and the description of the algorithm, as well as the proof of its validity.