Computing irreducible representations of finite groups

L{ászl{ó Babai, Lajos R{ónyai
1990 Mathematics of Computation  
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
more » ... e 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 .
doi:10.1090/s0025-5718-1990-1035925-1 fatcat:theuji7o7bd77fribcpi34apqe