An Algorithm for Decomposing Representations of Finite Groups Using Invariant Projections

Abstract

We describe an algorithm for decomposing permutation representations of finite groups over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projecting to these subspaces. This allows us to reduce the problem to solving systems of quadratic equations. The current implementation of the suggested algorithm allows us to split representations with dimensions up to hundreds of thousands. Computational examples are given.