Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ1,…,λd2) of size ℓd there exists k⩾1 such that the tensor square of the irreducible representation of the symmetric group Skℓd with respect to the rectangular partition (kℓ,…,kℓ) contains the irreducible representation corresponding to the stretched partition (kλ1,…,kλd2). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.