A new algorithm for irreducible decomposition of representations of finite groups

Vladimir V Kornyak

doi:10.1088/1742-6596/1194/1/012060

Abstract

An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a complete set of mutually orthogonal projectors. By expressing the projectors through the basis elements of the centralizer ring of the representation, the problem is reduced to solving systems of quadratic equations. The current implementation of the algorithm is able to split representations of dimensions up to hundreds of thousands. Examples of calculations are given.

Highlights

The decomposition of linear representations of groups into irreducible subrepresentations is one of the central problems of group theory and its applications in physics
The most effective algorithm for solving this problem is a Las Vegas type probabilistic algorithm, called MeatAxe [1]. This algorithm is based on the calculation of the characteristic polynomial of a randomly generated matrix of the representation
The MeatAxe algorithm played an important role in solving the problem of classifying finite simple groups, where it was applied to representations of groups in linear spaces over small finite fields, such as GF(2)

Summary

Introduction

The decomposition of linear representations of groups into irreducible subrepresentations is one of the central problems of group theory and its applications in physics. The most effective algorithm for solving this problem is a Las Vegas type probabilistic algorithm, called MeatAxe [1]. This algorithm is based on the calculation of the characteristic polynomial of a randomly generated matrix of the representation. The MeatAxe algorithm played an important role in solving the problem of classifying finite simple groups, where it was applied to representations of groups in linear spaces over small finite fields, such as GF(2). MeatAxe is inefficient in characteristic zero due to the rapid growth of numerical coefficients of characteristic polynomials with the matrix dimension, and due to the fact that in characteristic zero a random matrix with high probability has an irreducible characteristic polynomial

Objectives

Conclusion