We introduce an algorithm to decompose orthogonal matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a $d$-dimensional representation of $S_n$ is shown to have a complexity of $O(n^2 d^3)$ operations for determining which irreducible representations are present and their corresponding multiplicities and a further $O(n d^4)$ operations to fully decompose representations with non-trivial multiplicities. These complexity bounds are pessimistic and in a practical implementation using floating point arithmetic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on the problem of computing multiplicities of two tensor products of irreducible representations (the Kronecker coefficients problem) as well as higher order tensor products. For hook and hook-like irreducible representations the algorithm has polynomial complexity as $n$ increases. We also demonstrate an application to constructing a basis of multivariate orthogonal polynomials with respect to a tensor product weight so that applying a permutation of variables induces an irreducible representation.

翻译：我们提出一种算法，可将对称群在实数域上的正交矩阵表示分解为不可约表示，该算法作为副产品还能计算不可约表示的重数。该算法应用于$S_n$的$d$维表示时，确定存在的不可约表示及其对应重数的计算复杂度为$O(n^2 d^3)$次运算，而对具有非平凡重数的表示进行完全分解还需额外$O(n d^4)$次运算。这些复杂度界限是保守估计，在实际使用浮点运算并利用稀疏性的实现中，我们观察到更优的复杂度。我们通过在计算不可约表示的二重张量积（克罗内克系数问题）及高阶张量积的重数问题上演示该算法。对于钩形及类钩形不可约表示，该算法在$n$增大时具有多项式复杂度。我们还展示了该算法在构建关于张量积权重的多元正交多项式基方面的应用，使得变量置换作用可诱导出不可约表示。