# A new algorithm for irreducible decomposition of representations of finite groups

## TL;DR

This paper presents a novel algorithm for decomposing finite group representations into irreducible components, leveraging invariant inner products and centralizer rings, capable of handling large-dimensional cases.

## Contribution

The paper introduces a new algorithm that reduces the irreducible decomposition problem to solving quadratic equations using centralizer rings, improving scalability.

## Key findings

- Successfully decomposes representations with dimensions up to hundreds of thousands.
- Provides examples demonstrating the algorithm's effectiveness.
- Reduces the problem to solving systems of quadratic equations.

## 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.