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Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

  • Skip Garibaldi (a1) and Daniel K. Nakano (a2)


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.



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[AST] Andersen, H. H., Stroppel, C., and Tubbenhaur, D., Cellular structures using Uq-tilting modules. arxiv:1 503.0022
[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
[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
[BNP] Bendel, C. P., Nakano, D. K., and Pillen, C., Extensions for Frobenius kernels. J. Algebra 272(2004), no. 2, 476511.
[Bor] Borel, A., Linear algebraic groups. Second ed., Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.
[Bou Al] Bourbaki, N., Algebra I: Chapters 1-3. Elements of Mathematics, Springer-Verlag, Berlin,1989.
[Bou A4] Bourbaki, N., Algebra II: Chapters 4-7, Springer-Verlag, 1988.
[Bou A9] Bourbaki, N., Algèbre IX, Hermann, Paris, 1959.
[Bou L4] Bourbaki, N., Lie groups and Lie algebras: Chapters 4-6. Springer-Verlag, Berlin, 2002.
[Bou L7] Bourbaki, N., Lie groups and Lie algebras: Chapters 7-9. Springer-Verlag, Berlin, 2005.
[Brown] Brown, R. B., Groups of type E7. J. Reine Angew. Math. 236(1969), 79102.
[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
[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
[DS] Doty, S. R. and Sullivan, J. B., The submodule structure of Weyl modules for SL3. J. Algebra 96(1985), no. 1, 7893.
[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
[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.
[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.
[Ga] Garibaldi, S., Vanishing of trace forms in low characteristic. Algebra Number Theory 3(2009), no. 5, 543566.
[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
[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
[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.
[Groll] Grothendieck, A., Schémas en groupes III. Société Mathématique de France, 2011.
[GW09] Goodman, R. and Wallach, N. R., Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255, Springer, Dordrecht, 2009.
[Hiss] Hiss, G., Die adjungierten Darstellungen der Chevalley-Gruppen. Arch. Math. (Basel) 42(1984), no. 5, 408416.
[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
[Jan] Jantzen, J. C., Representations of algebraic groups. Second ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003.
[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.
[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.
[MR] Micali, A. and Revoy, Ph., Modules quadratiques. Bull. Soc. Math. France Mé m. (1979), no. 63,144 pp.
[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
[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.
[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.
[St] Steinberg, R., Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968.
[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
[W] Willems, W., Metrische G-Moduln über Körpern der Charakteristik 2. Math. Z. 157(1977), no. 2, 131139.
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Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

  • Skip Garibaldi (a1) and Daniel K. Nakano (a2)


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