Decomposing Linear Representations of Finite Groups

TL;DR

This paper introduces an open-source GAP package for decomposing finite group representations over complex numbers into irreducibles, filling a gap in available computational tools for characteristic zero cases.

Contribution

The authors develop and document a GAP package for characteristic zero representation decomposition, providing implementation details and demonstrating its application in semidefinite programming.

Findings

Provides a new open-source tool for complex representation decomposition

Details implementation strategies for efficient computation

Shows application in reducing semidefinite program sizes

Abstract

We develop a package using the computer algebra system GAP for computing the decomposition of a representation $\rho$ of a finite group $G$ over $\mathbb{C}$ into irreducibles, as well as the corresponding decomposition of the centraliser ring of $\rho(G)$. 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.