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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing irreducible representations of finite groups

Authors: László Babai and Lajos Rónyai
Journal: Math. Comp. 55 (1990), 705-722
MSC: Primary 68Q40; Secondary 11R33, 20C40
MathSciNet review: 1035925
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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 $ \mathcal{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 $ \mathcal{V}$ is given by the list of matrices $ \{ \mathcal{V}(g);g \in G\} $; and in randomized (Las Vegas) polynomial time under the more concise input $ \{ \mathcal{V}(g);g \in S\} $, where S is a set of generators of G.

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Article copyright: © Copyright 1990 American Mathematical Society