\textit{A. M. Garsia} and \textit{M. Haiman} [Proc. Natl. Acad. Sci. USA 90, No. 8, 3607-3610 (1993; Zbl 0831.05062)] have conjectured that the \(q,t\)-Macdonald polynomials have a representation theoretic interpretation in terms of the \(S_n\)-module spanned by the partial derivatives of \(\Delta_{\mu}({\mathbf x}; {\mathbf y}) = \det[x_i^{h_j} y_i^{k_j}]_{i,j=1}^n\) where \(\mu\) is a partition of \(n\). We order the cells in the Young diagram for \(\mu\) from left to right and then from the longest row to the shortest, and the \(j\)th square is in row \(h_j-1\) and column \(k_j-1\). The author proves that the span of the partial derivatives of \(\Delta_{\mu}\) is equal to the span of the translates of \(\Delta_{\mu}\), and then analyzes the decomposition of the translate \(\Delta_{\mu}[{\mathbf x} + {\mathbf x'}; {\mathbf y} + {\mathbf y'}]\) into irreducible isotypic components of the span of the partial derivatives. An algorithm for computing the coefficients and a multi-dimensional analog are also given.