On certain graded S n-modules and the q-Kostka polynomials

Abstract

We derive here a number of properties of the q-Kostka polynomials K λ, μ ( q). In particular we obtain a very accessible proof that these polynomials have non-negative integer coefficients. Other monotonicity properties are also derived. These results are obtained by studying certain graded S n -modules R μ which afford a character that may be expressed in terms of the K λ, μ ( q). Certain nesting properties of the R μ which correspond to the dominance order of partitions then translate themselves into combinatorial inequalities involving the K λ, μ ( q). The modules R μ have been given an elementary presentation by DeConcini and Procesi ( Invent. Math. 64 (1981), 203–219), as rather simple quotients of the polynomial ring Q[ x 1, x 2, ..., x n ]. We show here that their basic properties may also be derived in an entirely elementary manner.