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 multiplicities of irreducible representations 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.

翻译：我们引入了一种算法,将对称组对真实的正向矩阵表达方式分解成不可复制的表示方式,作为一种副产品,还计算了不可复制的表示方式的多重性。对以美元为单位的美元表示方式应用了一种算法,其复杂性为O(n%2 d%3)美元,用于确定不可复制的表示方式的多重性,以及另一个以美元为单位的用美元(n d%4)来完全拆解与非三重多重性的表示方式的表示方式。这些复杂界限是悲观的,而且是在使用浮动点算术和开发宽度的实际实施过程中,我们观察到了更好的复杂性。我们用这种算法证明了计算两种不可复制的表示方式(克朗系数问题)和高压产品的多重性问题。对于钩和钩式的不可复制性表示方式,算法具有以美元为单位的多重复杂性。