ON CERTAIN GRADED SN-MODULES AND THE Q-KOSTKA POLYNOMIALS

We derive here a number of properties of the q-Kostka polynomials Kλ,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 Sn-modules Rμ which afford a character that may be expressed in terms of the Kλ,μ(q). Certain nesting properties of the Rμ which correspond to the dominance order of partitions then translate themselves into combinatorial inequalities involving the Kλ,μ(q). The modules Rμ 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[x1, x2, ...,xn]. We show here that their basic properties may also be derived in an entirely elementary manner. © 1992.