Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition $({\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_{d^2})$ of size $\ell d$thereexists$k\ge 1$ such that the tensor square of the irreducible representation of the symmetric group $S_{k\mathrm{\ell }d}$ with respect to the rectangular partition $(k\mathrm{\ell },\mathrm{.}\mathrm{.}\mathrm{.},k\mathrm{\ell })$ contains the irreducible representation corresponding to the stretched partition $(k{\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},k{\lambda }_{d^2})$. 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.