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Incorporating permutation equivariance into neural networks has proven to be useful in ensuring that models respect symmetries that exist indata. Symmetric tensors, which naturally appearin statistics, machine learning, and graph theory,are essential for many applications in physics,chemistry, and materials science, amongst others. However, existing research on permutationequivariant models has not explored symmetrictensors as inputs, and most prior work on learningfrom these tensors has focused on equivarianceto Euclidean groups. In this paper, we presenttwo different characterisations of all linear permutation equivariant functions between symmetricpower spaces of $\mathbb{R}^{n}$. We show on two tasks thatthese functions are highly data efficient comparedto standard MLPs and have potential to generalisewell to symmetric tensors of different sizes.