Science Fiction and Macdonald's Polynomials

Abstract

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules ${\mathbf{M}}_{\mu }$ and corresponding elements of the Macdonald basis. We recall that ${\mathbf{M}}_{\mu }$ is defined for a partition $μ\part n$, as the linear span of derivatives of a certain bihomogeneous polynomial ${\Delta }_{\mu }(x,y)$ in the variables $x_1,x_2,...,x_n,y_1,y_2,...,y_n$. It has been conjectured by Garsia and Haiman that ${\mathbf{M}}_{\mu }$ has $n!$ dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial ${\widetilde{H}}_μ(x;q,t)=\sum_{λ\part n} S_λ(X) {\widetilde{K}}_{λμ}(q,t)$ where the ${\tilde{K}}_{\lambda \mu }(q,t)$ are related to the Macdonald $q,t$-Kostka coefficients $K_{\lambda \mu }(q,t)$ by the identity ${\tilde{K}}_{\lambda \mu }(q,t)=K_{\lambda \mu }(q,1/t)t^{n(\mu )}$ with $n(\mu )$ the x-degree of ${\Delta }_{\mu }(x;y)$. Computer data has suggested that as $\nu$ varies among the immediate predecessors of a partition $\mu$, the spaces ${\mathbf{M}}_{\nu }$ behave like a boolean lattice. We formulate a number of remarkable conjectures about the Macdonald polynomials. In particular we obtain a representation theoretical interpretation for some of the symmetries that can be found in the computed tables of $q,t$-Kostka coefficients.

47 pages, TeX