Hostname: page-component-77f85d65b8-zzw9c Total loading time: 0 Render date: 2026-04-22T21:57:11.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Spinors and essential dimension

Part of: Forms and linear algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 March 2017

Skip Garibaldi  and
Robert M. Guralnick
Skip Garibaldi
Affiliation:
Center for Communications Research, San Diego, CA 92121, USA email skip@member.ams.org
Robert M. Guralnick
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA email guralnic@usc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan et al. [Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), 533–544], and Chernousov and Merkurjev [Essential dimension of spinor and Clifford groups, Algebra Number Theory 8 (2014), 457–472] to fields of characteristic different from two. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.

Keywords

spin modulespin groupcharacteristic 2essential dimensiongenerically freehalf-spin representation

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups
Secondary: 11E88: Quadratic spaces; Clifford algebras 20G10: Cohomology theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 3 , March 2017 , pp. 535 - 556
Copyright
© The Authors 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.)

Article purchase

Temporarily unavailable

References

Andreev, E. M. and Popov, V. L., Stationary subgroups of points of general position in the representation space of a semisimple Lie group , Funct. Anal. Appl. 5 (1971), 265–271.Google Scholar
Aschbacher, M. and Seitz, G., Involutions in Chevalley groups over fields of even order , Nagoya Math. J. 63 (1976), 1–91.Google Scholar
Azad, H., Barry, M. and Seitz, G., On the structure of parabolic subgroups , Comm. Algebra 18 (1990), 551–562.Google Scholar
Baek, S. and Merkurjev, A., Essential dimension of central simple algebras , Acta Math. 209 (2012), 1–27.Google Scholar
Borel, A. and Springer, T. A., Rationality properties of linear algebraic groups , in Algebraic groups and discontinuous subgroups (Proceedings of Symposium in Pure Mathematics, Boulder, CO, 1965) (American Mathematical Society, Providence, RI, 1966), 26–32.Google Scholar
Bourbaki, N., Lie groups and Lie algebras (Springer, Berlin, 2002).Google Scholar
Brosnan, P., Reichstein, Z. and Vistoli, A., Essential dimension, spinor groups, and quadratic forms , Ann. of Math. (2) 171 (2010), 533–544.Google Scholar
Buhler, J. and Reichstein, Z., On the essential dimension of a finite group , Compositio Math. 106 (1997), 159–179.Google Scholar
Chernousov, V. and Merkurjev, A. S., Essential dimension of spinor and Clifford groups , Algebra Number Theory 8 (2014), 457–472.CrossRefGoogle Scholar
Chernousov, V. and Serre, J.-P., Lower bounds for essential dimensions via orthogonal representations , J. Algebra 305 (2006), 1055–1070.CrossRefGoogle Scholar
Chevalley, C., The algebraic theory of spinors (Springer, Berlin, 1997), reprint of the 1954 edition.Google Scholar
Collingwood, D. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebras (Van Nostrant Reinhold, New York, 1993).Google Scholar
Demazure, M. and Grothendieck, A., Schémas en groupes II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, vol. 152 (Springer, Berlin, 1970).Google Scholar
Florence, M., On the essential dimension of cyclic p-groups , Invent. Math. 171 (2008), 175–189.Google Scholar
Fulman, J., Guralnick, R. and Stanton, D., Asymptotics of the number of involutions in finite classical groups. Preprint (2016), arXiv:1602.03611.Google Scholar
Garibaldi, S., Isotropic trialitarian algebraic groups , J. Algebra 210 (1998), 385–418.Google Scholar
Garibaldi, S., Cohomological invariants: exceptional groups and spin groups, Memoirs American Mathematical Society, vol. 937 (American Mathematical Society, Providence, RI, 2009), with an appendix by Detlev W. Hoffmann.Google Scholar
Garibaldi, S. and Guralnick, R. M., Simple groups stabilizing polynomials , Forum Math.: Pi 3 (2015), e3 (41 pages), doi:10.1017/fmp.2015.3.Google Scholar
Garibaldi, S. and Guralnick, R. M., Essential dimension of algebraic groups, including bad characteristic , Arch. Math. 107 (2016), 101–119.Google Scholar
Gatti, V. and Viniberghi, E., Spinors of 13-dimensional space , Adv. Math. 30 (1978), 137–155.Google Scholar
Gille, P. and Reichstein, Z., A lower bound on the essential dimension of a connected linear group , Comment. Math. Helv. 84 (2009), 189–212.Google Scholar
Guerreiro, M., Exceptional representations of simple algebraic groups in prime characteristic, PhD thesis, University of Manchester (1997), arXiv:1210.6919.Google Scholar
Guralnick, R. M., Liebeck, M. W., Macpherson, D. and Seitz, G. M., Modules for algebraic groups with finitely many orbits on subspaces , J. Algebra 196 (1997), 211–250.Google Scholar
Igusa, J.-I., A classification of spinors up to dimension twelve , Amer. J. Math. 92 (1970), 997–1028.Google Scholar
Karpenko, N. and Merkurjev, A., Essential dimension of quadrics , Invent. Math. 153 (2003), 361–372.Google Scholar
Karpenko, N. and Merkurjev, A., Essential dimension of finite p-groups , Invent. Math. 172 (2008), 491–508.Google Scholar
Knus, M.-A., Merkurjev, A. S., Rost, M. and Tignol, J.-P., The book of involutions, Colloquium Publications, vol. 44 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Liebeck, M. W., The affine permutation groups of rank 3 , Proc. Lond. Math. Soc. (3) 54 (1987), 477–516.Google Scholar
Liebeck, M. and Seitz, G., Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys Monographs, vol. 180 (American Mathematical Society, Providence, RI, 2012).Google Scholar
Lötscher, R., A fiber dimension theorem for essential and canonical dimension , Compositio Math. 149 (2013), 148–174.Google Scholar
Lötscher, R., MacDonald, M., Meyer, A. and Reichstein, Z., Essential dimension of algebraic tori , J. Reine Angew. Math. 677 (2013), 1–13.Google Scholar
Merkurjev, A., Essential dimension, quadratic forms–algebra, arithmetic, and geometry, Contemporary Mathematics, vol. 493, eds Baeza, R., Chan, W. K., Hoffmann, D. W. and Schulze-Pillot, R. (American Mathematical Society, Providence, RI, 2009), 299–325.Google Scholar
Merkurjev, A., Essential p-dimension of PGL(p 2) , J. Amer. Math. Soc. 23 (2010), 693–712.Google Scholar
Merkurjev, A., Essential dimension: A survey , Transform. Groups 18 (2013), 415–481.Google Scholar
Merkurjev, A., Essential dimension , in Séminaire Bourbaki, Astérisque, vol. 380 (Société Mathématique de France, 2016), 423–448.Google Scholar
Merkurjev, A., Invariants of algebraic groups and retract rationality of classifying spaces , in Algebraic groups: structure and actions, Proceedings of Symposia in Pure Mathematics, vol. 94 (American Mathematical Society, Providence, RI, 2017). Preprint (2015), http://www.math.ucla.edu/∼merkurev/papers/retract-class-space.pdf.Google Scholar
Popov, A. M., Finite isotropy subgroups in general position of simple linear Lie groups , Trans. Moscow Math. Soc. (1988), 205–249 (Russian original: Trudy Moskov. Mat. Obschch. 50 (1987), 209–248, 262).Google Scholar
Popov, V. L., Classification of spinors of dimension 14 , Trans. Moscow Math. Soc. 37 (1980), 181–232.Google Scholar
Reichstein, Z., Essential dimension , in Proceedings of the international congress of mathematicians 2010 (World Scientific, Singapore, 2010).Google Scholar
Reichstein, Z. and Youssin, B., Essential dimensions of algebraic groups and a resolution theorem for G-varieties , Canad. J. Math. 52 (2000), 1018–1056, with an appendix by J. Kollár and E. Szabó.Google Scholar
Röhrle, G., On certain stabilizers in algebraic groups , Comm. Algebra 21 (1993), 1631–1644.Google Scholar
Rost, M., On 14-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, Preprint (1999), https://www.math.uni-bielefeld.de/∼rost/spin-14.html.Google Scholar
Rost, M., On the Galois cohomology of Spin(14), Preprint (1999), https://www.math.uni-bielefeld.de/∼rost/spin-14.html.Google Scholar
Steinberg, R., Lectures on Chevalley groups (Yale University, New Haven, CT, 1968).Google Scholar
Strade, H. and Farnsteiner, R., Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116 (Marcel Dekker, New York, 1988).Google Scholar