Linear groups and computation

TL;DR

This paper surveys methods and algorithms for practical computation with finitely generated linear groups over infinite fields, illustrating solutions to complex problems through computer experiments and discussing future research directions.

Contribution

It provides a comprehensive overview of computational techniques for linear groups over infinite fields, highlighting new algorithms and applications.

Findings

Demonstrated practical algorithms for linear group computations

Solved complex mathematical problems via computer experimentation

Discussed future research avenues in computational linear algebra

Abstract

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed. This article is aimed at a broad mathematical audience, and more particularly at users of group-theoretical methods and computer algebra systems.