Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

Garibaldi, Skip ; Nakano, Daniel K.

Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 395-421

Résumé

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.

Détail

DOI : 10.4153/CJM-2015-042-5

Mots-clés : 20G05, 11E39, 11E88, 15A63, 20G15, orthogonal representations, symmetric tensors, alternating forms, characteristic 2, splitreductive groups

Bibliographie

[AST] [AST] Andersen, H. H., Stroppel, C., and Tubbenhaur, D., Cellular structures using U-tilting modules. arxiv:1 503.0022 Google Scholar

[BaC] [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] [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] [BNP] Bendel, C. P., Nakano, D. K., and Pillen, C., Extensions for Frobenius kernels. J. Algebra 272(2004), no. 2, 476–511. Google Scholar | DOI

[Bor] [Bor] Borel, A., Linear algebraic groups. Second ed., Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. Google Scholar | DOI

[Bou Al] [Bou Al] Bourbaki, N., Algebra I: Chapters 1-3. Elements of Mathematics, Springer-Verlag, Berlin,1989. Google Scholar

[Bou A4] [Bou A4] Bourbaki, N., Algebra II: Chapters 4-7, Springer-Verlag, 1988. Google Scholar

[Bou A9] [Bou A9] Bourbaki, N., Algèbre IX, Hermann, Paris, 1959. Google Scholar

[Bou L4] [Bou L4] Bourbaki, N., Lie groups and Lie algebras: Chapters 4-6. Springer-Verlag, Berlin, 2002. Google Scholar

[Bou L7] [Bou L7] Bourbaki, N., Lie groups and Lie algebras: Chapters 7-9. Springer-Verlag, Berlin, 2005. Google Scholar

[Brown] [Brown] Brown, R. B., Groups of type E. J. Reine Angew. Math. 236(1969), 79–102. Google Scholar

[CS] [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 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] [DS] Doty, S. R. and Sullivan, J. B., The submodule structure of Weyl modules for SL. J. Algebra 96(1985), no. 1, 78–93. Google Scholar | DOI

[DV] [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] [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] [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] [Ga] Garibaldi, S., Vanishing of trace forms in low characteristic. Algebra Number Theory 3(2009), no. 5, 543–566. Google Scholar | DOI

[GN] [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] [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] [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. Google Scholar | DOI

[Groll] [Groll] Grothendieck, A., Schémas en groupes III. Société Mathématique de France, 2011. Google Scholar

[GW09] [GW09] Goodman, R. and Wallach, N. R., Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255, Springer, Dordrecht, 2009. Google Scholar | DOI

[Hiss] [Hiss] Hiss, G., Die adjungierten Darstellungen der Chevalley-Gruppen. Arch. Math. (Basel) 42(1984), no. 5, 408–416. Google Scholar | DOI

[HN] [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] [Jan] Jantzen, J. C., Representations of algebraic groups. Second ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003. Google Scholar

[KMRT] [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] 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] [MR] Micali, A. and Revoy, Ph., Modules quadratiques. Bull. Soc. Math. France Mé m. (1979), no. 63,144 pp. Google Scholar

[Se] [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] [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] [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] [St] Steinberg, R., Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968. Google Scholar

[SinW] [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] [W] Willems, W., Metrische G-Moduln über Körpern der Charakteristik 2. Math. Z. 157(1977), no. 2, 131–139. Google Scholar | DOI