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We introduce an algorithm to decompose orthogonal matrix representations ofthe symmetric group over the reals into irreducible representations, which as aby-product also computes the multiplicities of the irreducible representations.The algorithm applied to a $d$-dimensional representation of $S_n$ is shown tohave a complexity of $O(n^2 d^3)$ operations for determining multiplicities ofirreducible representations and a further $O(n d^4)$ operations to fullydecompose representations with non-trivial multiplicities. These complexitybounds are pessimistic and in a practical implementation using floating pointarithmetic and exploiting sparsity we observe better complexity. We demonstratethis algorithm on the problem of computing multiplicities of two tensorproducts of irreducible representations (the Kronecker coefficients problem) aswell as higher order tensor products. For hook and hook-like irreduciblerepresentations the algorithm has polynomial complexity as $n$ increases.

Representations of the symmetric group are decomposable in polynomial time