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Let $I$ be the ideal generated by alternating polynomials in two sets of $n$variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series ofthe graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by aminimal set of generators for $I$. In this paper we give simple upper bounds on$\text{dim}M_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees$(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For suchbi-degrees, we also find explicit bases for $M_{d_1, d_2}$. The main idea is todefine and study a nontrivial linear map from $M$ to a polynomial ring$\C[\rho_1, \rho_2,...]$.

$q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $(\C^2)^n$