Computational Bounds for Doing Harmonic Analysis on Permutation Modules of Finite Groups

TL;DR

This paper introduces a method to estimate the computational complexity of harmonic analysis on permutation modules of finite groups, leveraging orbital structures and irreducible submodule multiplicities, with notably efficient bounds for symmetric group actions.

Contribution

It presents a novel approach to bounding arithmetic operations in harmonic analysis on permutation modules using orbital structures and irreducible multiplicities.

Findings

Bounds are surprisingly small for symmetric group permutation modules.

The method effectively exploits orbital structure and irreducible submodule multiplicities.

Applicable to various finite group actions, especially symmetric groups.

Abstract

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.