  We prove that for any partition $(\lambda_1,...,\lambda_{d^2})$ of size $\ell
d$ there exists $k\ge 1$ such that the tensor square of the irreducible
representation of the symmetric group $S_{k\ell d}$ with respect to the
rectangular partition $(k\ell,...,k\ell)$ contains the irreducible
representation corresponding to the stretched partition
$(k\lambda_1,...,k\lambda_{d^2})$. We also prove a related approximate version
of this statement in which the stretching factor $k$ is effectively bounded in
terms of $d$. This investigation is motivated by questions of geometric
complexity theory.
