  We study generating functions of $\frac{1}{4}$-BPS states in $\mathcal{N}=4$
super Yang-Mills at finite $N$ by attempting to generalize the
Harish-Chandra-Itzykson-Zuber integral to multiple commuting matrices. This
allows us to compute the overlaps of two or more generating functions; such
calculations arise in the computation of two-point correlators in the
free-field limit. We discuss the four-matrix HCIZ integral in the $U(2)$
context and lay out a prescription for finding a more general formula for
$N>2$. We then discuss its connections with the restricted Schur polynomial
operator basis. Our results generalize readily to arbitrary numbers of
matrices, opening up the opportunity to study more generic BPS operators.
