We introduce an algorithm to decompose 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is shown to have a complexity of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {O}}(n^2 d^3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> <msup> <mi>d</mi> <mn>3</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> operations for determining which irreducible representations are present and their corresponding multiplicities and a further <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {O}}(n d^4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <msup> <mi>d</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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 homogeneous polynomials so that applying a permutation of variables induces an irreducible representation.

Representations of the Symmetric Group are Decomposable in Polynomial Time

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2025年2月10日

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Sheehan Olver

摘要

We introduce an algorithm to decompose 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 $S_n$ $S_{n}$ is shown to have a complexity of ${\mathcal {O}}(n^2 d^3)$ $O (n^{2} d^{3})$ operations for determining which irreducible representations are present and their corresponding multiplicities and a further ${\mathcal {O}}(n d^4)$ $O (n d^{4})$ 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 homogeneous polynomials so that applying a permutation of variables induces an irreducible representation.