Science Fiction and Macdonald's Polynomials

Abstract

This work studies the remarkable relationships that hold among certain m-tuples of the Garsia-Haiman modules $ {\bf M}_\mu$ and corresponding elements of the Macdonald basis. We recall that ${\bf M}_\mu$ is defined for a partition $\mu\part n$, 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 ${\bf M}_\mu$ has $n!$ dimensions and that its bigraded Frobenius characteristic is given by the symmetric polynomial ${\widetilde{H}}_\mu(x;q,t)=\sum_{\lambda\part n} S_\lambda(X) {\widetilde{K}}_{\lambda\mu}(q,t)$ where the ${\widetilde{K}}_{\lambda\mu}(q,t)$ are related to the Macdonald $q,t$-Kostka coefficients $ K_{\lambda\mu}(q,t)$ by the identity ${\widetilde{K}}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu)}$ 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 ${\bf M}_\nu$ 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.