SOME NATURAL BIGRADED Sn-MODULES and q,t-KOSTKA COEFFICIENTS

We construct for each μ ` n a bigraded Sn-module Hμ and conjecture that its Frobenius characteristic Cμ(x; q, t) yields the Macdonald coefficients Kλμ(q, t). To be precise, we conjecture that the expansion of Cμ(x; q, t) in terms of the Schur basis yields coefficients Cλμ(q, t) which are related to the Kλμ(q, t) by the identity Cλμ(q, t) = Kλμ(q, 1/t)t. The validity of this would give a representation theoretical setting for the Macdonald basis {Pμ(x; q, t)}μ and establish the Macdonald conjecture that the Kλμ(q, t) are polynomials with positive integer coefficients. The space Hμ is defined as the linear span of derivatives of a certain bihomogeneous polynomial ∆μ(x, y) in the variables x1, x2, . . . , xn, y1, y2, . . . , yn. On the validity of our conjecture Hμ would necessarily have n! dimension. We refer to the latter assertion as the n!-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials Cλμ(q, t) have a number of basic properties in common with the coefficients K̃λμ(q, t) = Kλμ(q, 1/t)t. For instance, we show that Cλμ(0, t) = K̃λμ(0, t), Cλμ(q, 0) = K̃λμ(q, 0) and show that on the n! conjecture we must also have the equalities Cλμ(1, t) = K̃λμ(1, t) and Cλμ(q, 1) = K̃λμ(q, 1). The conjectured equality Cλμ(q, t) = Kλμ(q, 1/t)t will be shown here to hold true when λ or μ is a hook. It has also been shown (see [9]) when μ is a 2-row or 2-column partition and in [18] when μ is an augmented hook. Introduction Throughout this writing μ will be a partition of n and μ′ will denote its conjugate. We shall also identify μ with its Ferrers’ diagram. As customary, for μ = (μ1 ≥ μ2 ≥ · · · ≥ μk > 0), we let