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Representations of the Symmetric Group are Decomposable in Polynomial Time.

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Title: Representations of the Symmetric Group are Decomposable in Polynomial Time.
Authors: Olver, Sheehan1 (AUTHOR) s.olver@imperial.ac.uk
Source: Foundations of Computational Mathematics. Apr2026, Vol. 26 Issue 2, p1069-1092. 24p.
Subjects: Representation theory, Polynomial time algorithms, Matrix decomposition, Tensor products, Algorithms, Symmetry groups
Abstract: 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 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 homogeneous polynomials so that applying a permutation of variables induces an irreducible representation. [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: 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 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 homogeneous polynomials so that applying a permutation of variables induces an irreducible representation. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10208-025-09697-8
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      – SubjectFull: Matrix decomposition
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      – SubjectFull: Tensor products
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      – SubjectFull: Symmetry groups
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              Text: Apr2026
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