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Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

Published online by Cambridge University Press:  20 November 2018

Skip Garibaldi  and
Daniel K. Nakano
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
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Abstract

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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.

Keywords

20G0511E3911E8815A6320G15orthogonal representationssymmetric tensorsalternating formscharacteristic 2split reductive groups
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 2 , 01 April 2016 , pp. 395 - 421
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), 37–63. doi=10.2140/pjm.2015.279.37 Google Scholar
[BC] Benson, D. J. and Carlson, J. F., Diagrammatic methods for modular representations and cohomology. Comm. Algebra 15(1987), no. 1-2, 53–121.http://dx.doi.Org/10.1080/00927878708823414 Google Scholar
[BNP] Bendel, C. P., Nakano, D. K., and Pillen, C., Extensions for Frobenius kernels. J. Algebra 272(2004), no. 2, 476–511.http://dx.doi.org/10.1016/j.jalgebra.2OO3.04.003 Google 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), 79–102.Google Scholar
[CS] Chernousov, V. and Serre, J-P., Lower bounds for essential dimensions via orthogonal representations. J. Algebra 305(2006), 1055–1070. http://dx.doi.Org/10.1016/j.jalgebra.2005.10.032 Google Scholar
[De B] De Bruyn, B., On the Grassmann modules for the symplectic groups. J. Algebra 324(2010), 218–230.http://dx.doi.Org/10.1016/j.jalgebra.2O10.03.033 Google Scholar
[DS] Doty, S. R. and Sullivan, J. B., The submodule structure of Weyl modules for SL3. J. Algebra 96(1985), no. 1, 78–93.http://dx.doi.org/10.1016/0021-8693(85)90040-7 Google Scholar
[DV] Drápal, A. and Vojtěchovský, P., Symmetric multilinear forms and polarization of polynomials. Linear Algebra Appl. 431(2009), no. 5-7, 998–1012.http://dx.doi.Org/10.1016/j.laa.2009.03.052 Google 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), 39–166.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, 543–566.http://dx.doi.org/10.2140/ant.20093.543 Google Scholar
[GN] Gross, B. H. and Nebe, G., Globally maximal arithmetic groups. J. Algebra 272(2004), no. 2, 625–642.http://dx.doi.Org/10.1016/j.jalgebra.2003.09.033 Google 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, 60–89.http://dx.doi.Org/10.1006/jabr.1999.7943 Google 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), 1067–1081.http://dx.doi.org/10.1006/jabr.1995.1227 Google 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, 408–416.http://dx.doi.org/10.1007/BF01190689 Google Scholar
[HN] Hemmer, D. J. and Nakano, D. K., On the cohomology of Specht modules. J. Algebra 306(2006), no. 1, 191–200.http://dx.doi.Org/10.1016/j.jalgebra.2006.03.044 Google 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), 143–174.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, 225–274.http://dx.doi.Org/10.1016/0001-8708(77)90041-X Google 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. 167–266.Google 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), 45–59.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, 131–139. http://dx.doi.org/10.1007/BF01215147 Google Scholar