Nonvanishing of Kronecker coefficients for rectangular shapes

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We prove that for any partition (λ₁,...,λd₂) of size ℓ d there exists k≥ 1 such that the tensor square of the irreducible representation of the symmetric group Sₖℓ d 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. This investigation is motivated by questions of geometric complexity theory.