An integral ‘Meat-axe’

Summary

Abstract

We describe algorithms for working with matrices of integers, performing most of the Meataxe functions on them. In particular, we can chop integral representations of groups into irreducibles, and prove irreducibility.

Introduction

This paper describes some algorithms sufficient to construct and work with representations of finite groups over the ordinary integers. The methods are, however, of quite general use when working with matrices of integers.

In general, the algorithms and programs closely match those used for working with matrices over finite fields, but several new problems arise.

The first problem is to find an effective replacement for Gaussian elimination—the workhorse method for finite fields. The replacement described here, called “The Module Handler”, although of restricted functionality, seems to do the job, and much of the work of this paper consists of doing such jobs as “Nullspace” and “Find a ℤ-basis” using only this functionality.

Another problem is to work with integers of unbounded size, and yet to prevent the size growing too much. I have some hopes that the methods described here are quite good in this respect, but have made no serious effort to analyse their behaviour.

A third “problem” that I expected is that matrices of non-zero nullity are vanishingly rare in characteristic zero. Astonishingly, in my work so far I have not encountered this problem at all. We have

Research Problem 1Understand why “small” combinations of group elements often have non-zero nullity.

In section 2, the concept of a Module Handler is defined and introduced. In section 3, an implementation based on p-adic lift is described. In section 4, the main programs that use the module handler are described.