Some Natural Bigraded Sₙ-Modules

We construct for each μ n a bigraded Sₙ-module H_μ 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)tn(μ ). The validity of this would give a representation theoretical setting for the Macdonald basis \ P_μ(x;q,t)_μ and establish the Macdonald conjecture that the Kλμ(q,t) are polynomials with positive integer coefficients. The space H_μ is defined as the linear span of derivatives of a certain bihomogeneous polynomial Δ_μ(x,y) in the variables x₁,x₂,… ,xₙ, y₁,y₂,… ,yₙ. On the validity of our conjecture H_μ 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)tn(μ ). 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)tn(μ ) 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.