Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T21:51:36.957Z Has data issue: false hasContentIssue false

Explicit tilting complexes for the Broué conjecture on 3-blocks

Published online by Cambridge University Press:  07 May 2010

Ayala Bar-Ilan
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Tzviya Berrebi
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Genadi Chereshnya
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Ruth Leabovich
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Mikhal Cohen
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
Mary Schaps
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
Get access

Summary

Abstract

The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks, mostly with defect group C3 × C3 or C5 × C5. In this paper, we exhibit explicit tilting complexes from the Brauer correspondent to the global block B for a number of Morita equivalence classes of blocks of defect group C3 × C3. We also describe a database with data sheets for over a thousand blocks of abelian defect group in the ATLAS group and their subgroups.

Introduction

Let G be a finite group and let k be a field of characteristic p, where p divides |G|. Let kG =⊕ Bi be a decomposition of the group algebra into blocks, and let Di be the defect group of the block Bi, of order. By Brauer's Main Theorems (see [1] for an accessible exposition) there is a one-to-one correspondence between blocks of kG with defect group Di and blocks of kNG(Di) with defect group Di. Let bi be the block corresponding to Bi, called its Brauer correspondent.

Broué [5] has conjectured that if Di is abelian and Bi is a principal block, then Bi and bi are derived equivalent, i.e., the bounded derived categories Db(Bi) and Db(bi) are equivalent. In fact, it is generally believed by researchers in the field that the hypothesis that Bi be principal is unnecessary.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2007

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.

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.

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.

Available formats
×