Computing irreducible representations of finite groups

Abstract

The bit complexity of computing irreducible representations of finite groups is considered. Exact computations in algebraic number fields are performed symbolically. A polynomial-time algorithm for finding a complete set of inequivalent irreducible representations over the field of complex numbers of a finite group given by its multiplication table is presented. It follows that some representative of each equivalence class of irreducible representations admits a polynomial-size description. The problem of decomposing a given representation V of the finite group G over an algebraic number field F into absolutely irreducible constituents is considered. It is shown that this can be done in deterministic polynomial time if V is given by the list of matrices (V(g); g in G) and in randomized (Las Vegas) polynomial time under the more concise input (V(g); g in S), where S is a set of generators of G.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>