Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-14T06:29:36.713Z Has data issue: false hasContentIssue false

New computations in the Monster

Published online by Cambridge University Press:  06 July 2010

Robert A. Wilson
Affiliation:
School of Mathematical Sciences Queen Mary, University of London Mile End Road, London E1 4NS
James Lepowsky
Affiliation:
Rutgers University, New Jersey
John McKay
Affiliation:
Concordia University, Montréal
Michael P. Tuite
Affiliation:
National University of Ireland, Galway
Get access

Summary

Abstract

We survey recent computational results concerning the Monster sporadic simple group. The main results are: progress towards a complete classification of the maximal subgroups, including showing that L2(27) is not a subgroup; showing that the 196882-dimensional module over GF(2) supports a quadratic form; a complete set of explicit conjugacy class representatives; small representations of most of the maximal subgroups; and a partial classification of the ‘nets’ (in the sense of Norton).

Introduction

Our aim in this paper is to update the survey by describing the various explicit computations which have been performed in the Monster group, and the new information about the Monster which has resulted from these calculations. We begin by summarising for the benefit of readers who do not have that paper to hand.

The smallest matrix representations of the Monster have dimension 196882 in characteristics 2 and 3, and dimension 196883 in all other characteristics. Three of these representations (over the fields of orders 2, 3, and 7) are now available explicitly. It is hoped that the data and programs to manipulate them will be made available in the next release of Magma. The generating matrices are stored in a compact way, and never written out in full. The basic operation of the system is to calculate the action of a generator on a vector of the underlying module.

Our first construction was carried out over the field GF(2) of two elements in the interests of speed, and proceeded by amalgamating various 3-local subgroups.

Type
Chapter
Information
Moonshine - The First Quarter Century and Beyond
Proceedings of a Workshop on the Moonshine Conjectures and Vertex Algebras
, pp. 393 - 403
Publisher: Cambridge University Press
Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • New computations in the Monster
    • By Robert A. Wilson, School of Mathematical Sciences Queen Mary, University of London Mile End Road, London E1 4NS
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.019
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • New computations in the Monster
    • By Robert A. Wilson, School of Mathematical Sciences Queen Mary, University of London Mile End Road, London E1 4NS
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.019
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • New computations in the Monster
    • By Robert A. Wilson, School of Mathematical Sciences Queen Mary, University of London Mile End Road, London E1 4NS
  • Edited by James Lepowsky, Rutgers University, New Jersey, John McKay, Concordia University, Montréal, Michael P. Tuite, National University of Ireland, Galway
  • Book: Moonshine - The First Quarter Century and Beyond
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730054.019
Available formats
×