### Abstract

We consider the bit-complexity of the problem stated in the title. Exact computations in algebraic number fields are performed symbolically. We present a polynomial-time algorithm to find a complete set of nonequivalent irreducible representations over the field of complex numbers of a finite group given by its multiplication table. In particular, it follows that some representative of each equivalence class of irreducible representations admits a polynomial-size description. We also consider the problem of decomposing a given representation 'V of the finite group G over an algebraic number field F into absolutely irreducible constituents. We are able to do this in deterministic polynomial time if 'V is given by the list of matrices (^(g) ', g 6 G); and in randomized (Las Vegas) polynomial time under the more concise input ('P'(g); g € S), where S is a set of generators of G.

Original language | English |
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Pages (from-to) | 705-722 |

Number of pages | 18 |

Journal | Mathematics of Computation |

Volume | 55 |

Issue number | 192 |

DOIs | |

Publication status | Published - Oct 1990 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Mathematics of Computation*,

*55*(192), 705-722. https://doi.org/10.1090/S0025-5718-1990-1035925-1