Skip to main content Accessibility help
×
Home
Hostname: page-component-dc8c957cd-pt62b Total loading time: 0.321 Render date: 2022-01-26T18:09:34.434Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Maximal Weight Composition Factors for Weyl Modules

Published online by Cambridge University Press:  20 November 2018

Jens Carsten Jantzen*
Affiliation:
Institut for Matematik, Aarhus Universitet, Ny Munkegade 118, DK-8000 Aarhus C, Denmark e-mail: jantzen@math.au.dk

Abstract

Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group ${{G}_{k}}$ over $k$ with root system $k$ . One associates with any dominant weight $\lambda $ for $R$ two ${{G}_{k}}$ -modules with highest weight $\lambda $ , the Weyl module $V{{(\lambda )}_{k}}$ and its simple quotient $V{{(\lambda )}_{k}}$ . Let $\lambda $ and $\mu $ be dominant weights with $\mu <\lambda $ such that $\mu $ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L{{(\mu )}_{k}}$ is a composition factor of $V{{(\lambda )}_{k}}$ , and they exhibit an example in type ${{E}_{8}}$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs $(\lambda ,\mu )$ , and another that relies only on the classiûcation of root systems.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

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.)

References

[1] Bourbaki, N., Groupes et algebres deLie: Chapitres 4, 5 et 6. Hermann, Paris, 1968Google Scholar
[2] Bourbaki, N., Groupes et algebres de Lie: Chapitres 7 et 8. Hermann, Paris 1975Google Scholar
[3] Cartan, E., Les groupes projectifs qui ne laissent invariante aucune multiplicity plane. Bull. Soc. Math. France 41(1913), 53 - 96. Google Scholar
[4] Freudenthal, H., Zur Berechnung der Charaktere der halbeinfachen Lieschen Gruppen II. Indag. Math. 16(1954), 487491. http://dx.doi.org/1 0.101 6/S1385-7258(54)50046-6 CrossRefGoogle Scholar
[5] Garibaldi, S., Guralnick, R. M., and Nakano, D. K., Globally irreducible Weyl modules. arxiv:1 604.08911 Google Scholar
[6] Humphreys, J., Introduction to Lie algebras and Representation Theory. Graduate Texts in Mathematics, 9, Springer, New York, 1972. Google Scholar
[7] Jantzen, J. C., Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen. Bonn. Math. Schr. 67(1973).Google Scholar
[8] Jantzen, J. C., Darstellungen halbeinfacher Gruppen und kontravariante Formen. J. reine angew. Math. 290(1977), 117141.Google Scholar
[9] Jantzen, J. C., Representations of algebraic groups. Second ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003. Google Scholar
[10] Veldkamp, F. D., Representations of algebraic groups oftype F4 in characteristic 2. J. Algebra 16(1970), 326339. http://dx.doi.org/!0.101 6/0021-8693(70)90013-X Google Scholar
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

Maximal Weight Composition Factors for Weyl Modules
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and 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 <service> account. Find out more about sending content to Dropbox.

Maximal Weight Composition Factors for Weyl Modules
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and 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 <service> account. Find out more about sending content to Google Drive.

Maximal Weight Composition Factors for Weyl Modules
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *