Skip to main content Accessibility help
×
Home
Hostname: page-component-568f69f84b-ftpnm Total loading time: 0.16 Render date: 2021-09-17T15:48:13.534Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

Published online by Cambridge University Press:  20 November 2018

Skip Garibaldi
Affiliation:
Institute for Pure and Applied Mathematics, UCLA, 460 Portola Plaza, Box 957121, Los Angeles, CA 90095-7121, USA and Center for Communications Research, San Diego, CA 92121, USA e-mail: skip@garibaldibros.com
Daniel K. Nakano
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA e-mail: nakano@uga.edu
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

References

[AST] Andersen, H. H., Stroppel, C., and Tubbenhaur, D., Cellular structures using Uq-tilting modules. arxiv:1 503.0022 Google Scholar
[BaC] Babic, A. and Chernousov, V., Lower bounds for essential dimensions in characteristic 2 via orthogonal representations. Pacific J. Math. 279(2015), 3763. doi=10.2140/pjm.2015.279.37 CrossRefGoogle Scholar
[BC] Benson, D. J. and Carlson, J. F., Diagrammatic methods for modular representations and cohomology. Comm. Algebra 15(1987), no. 1-2, 53121.http://dx.doi.Org/10.1080/00927878708823414 CrossRefGoogle Scholar
[BNP] Bendel, C. P., Nakano, D. K., and Pillen, C., Extensions for Frobenius kernels. J. Algebra 272(2004), no. 2, 476511.http://dx.doi.org/10.1016/j.jalgebra.2OO3.04.003 CrossRefGoogle Scholar
[Bor] Borel, A., Linear algebraic groups. Second ed., Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. http://dx.doi.org/10.1007/978-1-4612-0941-6 Google Scholar
[Bou Al] Bourbaki, N., Algebra I: Chapters 1-3. Elements of Mathematics, Springer-Verlag, Berlin,1989.Google Scholar
[Bou A4] Bourbaki, N., Algebra II: Chapters 4-7, Springer-Verlag, 1988.Google Scholar
[Bou A9] Bourbaki, N., Algèbre IX, Hermann, Paris, 1959.Google Scholar
[Bou L4] Bourbaki, N., Lie groups and Lie algebras: Chapters 4-6. Springer-Verlag, Berlin, 2002.Google Scholar
[Bou L7] Bourbaki, N., Lie groups and Lie algebras: Chapters 7-9. Springer-Verlag, Berlin, 2005.Google Scholar
[Brown] Brown, R. B., Groups of type E7. J. Reine Angew. Math. 236(1969), 79102.Google Scholar
[CS] Chernousov, V. and Serre, J-P., Lower bounds for essential dimensions via orthogonal representations. J. Algebra 305(2006), 10551070. http://dx.doi.Org/10.1016/j.jalgebra.2005.10.032 CrossRefGoogle Scholar
[De B] De Bruyn, B., On the Grassmann modules for the symplectic groups. J. Algebra 324(2010), 218230.http://dx.doi.Org/10.1016/j.jalgebra.2O10.03.033 CrossRefGoogle Scholar
[DS] Doty, S. R. and Sullivan, J. B., The submodule structure of Weyl modules for SL3. J. Algebra 96(1985), no. 1, 7893.http://dx.doi.org/10.1016/0021-8693(85)90040-7 CrossRefGoogle Scholar
[DV] Drápal, A. and Vojtěchovský, P., Symmetric multilinear forms and polarization of polynomials. Linear Algebra Appl. 431(2009), no. 5-7, 9981012.http://dx.doi.Org/10.1016/j.laa.2009.03.052 CrossRefGoogle Scholar
[Dy] Dynkin, E. B., Maximal subgroups of the classical groups. Amer. Math. Soc. Transi. (2) 6(1957), 245–378 ; (Russian) Trudy Moskov. Mat. Obšč. 1(1952), 39166.Google Scholar
[EKM] Elman, R. S., Karpenko, N., and Merkur jev, A., The algebraic and geometric theory of quadratic forms. American Mathematical Society Colloquium Publications, 56, American Mathematical Society, Providence, RI, 2008.Google Scholar
[Ga] Garibaldi, S., Vanishing of trace forms in low characteristic. Algebra Number Theory 3(2009), no. 5, 543566.http://dx.doi.org/10.2140/ant.20093.543 CrossRefGoogle Scholar
[GN] Gross, B. H. and Nebe, G., Globally maximal arithmetic groups. J. Algebra 272(2004), no. 2, 625642.http://dx.doi.Org/10.1016/j.jalgebra.2003.09.033 CrossRefGoogle Scholar
[GowK] Gow, R. and Kleshchev, A., Connections between representations of the symmetric group and the symplectic group in characteristic 2. J. Algebra 221(1999), no. 1, 6089.http://dx.doi.Org/10.1006/jabr.1999.7943 CrossRefGoogle Scholar
[GowW] Gow, R. and Willems, W., Methods to decide if simple self-dual modules over fields of characteristic 2 are of quadratic type. J. Algebra 175(1995), 10671081.http://dx.doi.org/10.1006/jabr.1995.1227 CrossRefGoogle Scholar
[Groll] Grothendieck, A., Schémas en groupes III. Société Mathématique de France, 2011.Google Scholar
[GW09] Goodman, R. and Wallach, N. R., Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255, Springer, Dordrecht, 2009.http://dx.doi.org/10.1007/978-0-387-79852-3 Google Scholar
[Hiss] Hiss, G., Die adjungierten Darstellungen der Chevalley-Gruppen. Arch. Math. (Basel) 42(1984), no. 5, 408416.http://dx.doi.org/10.1007/BF01190689 CrossRefGoogle Scholar
[HN] Hemmer, D. J. and Nakano, D. K., On the cohomology of Specht modules. J. Algebra 306(2006), no. 1, 191200.http://dx.doi.Org/10.1016/j.jalgebra.2006.03.044 CrossRefGoogle Scholar
[Jan] Jantzen, J. C., Representations of algebraic groups. Second ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
[KMRT] Knus, M.-A., Merkurjev, A. S., Rost, M., and Tignol, J.-P., The book of involutions. American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998.Google Scholar
[Mal] Mal'cev, A. I., On semi-simple subgroups of Lie groups. Amer. Math. Soc. Translation 9(1950), no. 33, 43; (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 8(1944), 143174.Google Scholar
[MR] Micali, A. and Revoy, Ph., Modules quadratiques. Bull. Soc. Math. France Mé m. (1979), no. 63,144 pp.Google Scholar
[Se] Seshadri, C. S., Geometric reductivity over arbitrary base. Advances in Math. 26(1977), no. 3, 225274.http://dx.doi.Org/10.1016/0001-8708(77)90041-X CrossRefGoogle Scholar
[SF] Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations. Monographs and Textbooks in Pure and Applied Mathematics, 116, Marcel Dekker, New York, 1988.Google Scholar
[SpSt] Springer, T. A. and Steinberg, R., Conjugacy classes. Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, 131, Springer, Berlin, 1970, pp. 167266.CrossRefGoogle Scholar
[St] Steinberg, R., Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968.Google Scholar
[SinW] Sin, P. and Willems, W., G-invariant quadratic forms. J. Reine Angew. Math. 420(1991), 4559.http://dx.doi.Org/10.1515/crll.1991.420.45 Google Scholar
[W] Willems, W., Metrische G-Moduln über Körpern der Charakteristik 2. Math. Z. 157(1977), no. 2, 131139. http://dx.doi.org/10.1007/BF01215147 CrossRefGoogle Scholar
You have Access
3
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.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
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.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
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.

Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
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? *