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Motivated by the Saxl conjecture and the tensor square conjecture, whichstates that the tensor squares of certain irreducible representations of thesymmetric group contain all irreducible representations, we study the tensorsquares of irreducible representations associated with square Young diagrams.We give a formula for computing Kronecker coefficients, which are indexed bytwo square partitions and a three-row partition, specifically one with a shortsecond row and the smallest part equal to 1. We also prove the positivity ofsquare Kronecker coefficients for particular families of partitions, includingthree-row partitions and near-hooks.

On the Kronecker product of Schur functions of square shapes