Computing irreducible representations of finite groups

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.