  We compute the auto-correlations functions of order $m\ge 1$ for the
characteristic polynomials of random matrices from certain subgroups of the
unitary groups $\U(2)$ and $\U(3)$ by applying branching rules. These subgroups
can be understood as analogs of Sato--Tate groups of $\USp(4)$ in our previous
paper. This computation yields symmetric polynomial identities with
$m$-variables involving irreducible characters of $\U(m)$ for all $m \ge 1$ in
an explicit, uniform way.
