  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.
