Decomposing Linear Representations of Finite Groups

Abstract

We develop a package using the computer algebra system GAP for computing the decomposition of a representation $ρ$ of a finite group $G$ over $\mathbb{C}$ into irreducibles, as well as the corresponding decomposition of the centraliser ring of $ρ(G)$. Currently, the only open-source programs for decomposing representations are for non-zero characteristic fields. While methods for characteristic zero are known, there are no open-source computer programs that implement these methods, nor are details on how to achieve good performance of such an implementation published. We aim to record such details and demonstrate an application of our program in reducing the size of semidefinite programs.