  We present a comprehensive analysis of an algorithm for evaluating
high-dimensional polynomials that are invariant under permutations and
rotations. The key bottleneck is the contraction of a high-dimensional
symmetric and sparse tensor with a specific sparsity pattern that is directly
related to the symmetries imposed on the polynomial. We propose an explicit
construction of a recursive evaluation strategy and show that it is optimal in
the limit of infinite polynomial degree.
