An Algorithm to Decompose Permutation Representations of Finite Groups: Polynomial Algebra Approach

TL;DR

This paper presents an algorithm that decomposes permutation representations of finite groups into irreducible components using polynomial algebra and Gr"obner bases, enabling analysis of large representations.

Contribution

The paper introduces a novel polynomial algebra-based algorithm for decomposing permutation representations of finite groups into irreducible parts.

Findings

Successfully decomposes large permutation representations.

Uses Gr"obner bases for polynomial ideal calculations.

Preliminary implementation handles representations up to tens of thousands in dimension.

Abstract

We describe an algorithm for splitting permutation representations of finite group 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 projection into these subspaces. An important element of the algorithm is the calculation of Gr\"obner bases of polynomial ideals. A preliminary implementation of the algorithm splits representations up to dimensions of tens of thousands. Some examples of computations are given in appendix.