Skip to main content Accessibility help

Spinors and essential dimension

  • Skip Garibaldi (a1) and Robert M. Guralnick (a2)


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.



Hide All
[AP71] 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), 265271.
[AS76] Aschbacher, M. and Seitz, G., Involutions in Chevalley groups over fields of even order , Nagoya Math. J. 63 (1976), 191.
[ABS90] Azad, H., Barry, M. and Seitz, G., On the structure of parabolic subgroups , Comm. Algebra 18 (1990), 551562.
[BM12] Baek, S. and Merkurjev, A., Essential dimension of central simple algebras , Acta Math. 209 (2012), 127.
[BS66] 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), 2632.
[Bou02] Bourbaki, N., Lie groups and Lie algebras (Springer, Berlin, 2002).
[BRV10] Brosnan, P., Reichstein, Z. and Vistoli, A., Essential dimension, spinor groups, and quadratic forms , Ann. of Math. (2) 171 (2010), 533544.
[BR97] Buhler, J. and Reichstein, Z., On the essential dimension of a finite group , Compositio Math. 106 (1997), 159179.
[CM14] Chernousov, V. and Merkurjev, A. S., Essential dimension of spinor and Clifford groups , Algebra Number Theory 8 (2014), 457472.
[CS06] Chernousov, V. and Serre, J.-P., Lower bounds for essential dimensions via orthogonal representations , J. Algebra 305 (2006), 10551070.
[Che97] Chevalley, C., The algebraic theory of spinors (Springer, Berlin, 1997), reprint of the 1954 edition.
[CM93] Collingwood, D. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebras (Van Nostrant Reinhold, New York, 1993).
[DG70] 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).
[Flo08] Florence, M., On the essential dimension of cyclic p-groups , Invent. Math. 171 (2008), 175189.
[FGS16] Fulman, J., Guralnick, R. and Stanton, D., Asymptotics of the number of involutions in finite classical groups. Preprint (2016), arXiv:1602.03611.
[Gar98] Garibaldi, S., Isotropic trialitarian algebraic groups , J. Algebra 210 (1998), 385418.
[Gar09] 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.
[GG15] Garibaldi, S. and Guralnick, R. M., Simple groups stabilizing polynomials , Forum Math.: Pi 3 (2015), e3 (41 pages), doi:10.1017/fmp.2015.3.
[GG16] Garibaldi, S. and Guralnick, R. M., Essential dimension of algebraic groups, including bad characteristic , Arch. Math. 107 (2016), 101119.
[GV78] Gatti, V. and Viniberghi, E., Spinors of 13-dimensional space , Adv. Math. 30 (1978), 137155.
[GR09] Gille, P. and Reichstein, Z., A lower bound on the essential dimension of a connected linear group , Comment. Math. Helv. 84 (2009), 189212.
[Gue97] Guerreiro, M., Exceptional representations of simple algebraic groups in prime characteristic, PhD thesis, University of Manchester (1997), arXiv:1210.6919.
[GLMS97] 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), 211250.
[Igu70] Igusa, J.-I., A classification of spinors up to dimension twelve , Amer. J. Math. 92 (1970), 9971028.
[KM03] Karpenko, N. and Merkurjev, A., Essential dimension of quadrics , Invent. Math. 153 (2003), 361372.
[KM08] Karpenko, N. and Merkurjev, A., Essential dimension of finite p-groups , Invent. Math. 172 (2008), 491508.
[KMRT98] 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).
[Lie87] Liebeck, M. W., The affine permutation groups of rank 3 , Proc. Lond. Math. Soc. (3) 54 (1987), 477516.
[LS12] 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).
[Lot13] Lötscher, R., A fiber dimension theorem for essential and canonical dimension , Compositio Math. 149 (2013), 148174.
[LMMR13] Lötscher, R., MacDonald, M., Meyer, A. and Reichstein, Z., Essential dimension of algebraic tori , J. Reine Angew. Math. 677 (2013), 113.
[Mer09] 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), 299325.
[Mer10] Merkurjev, A., Essential p-dimension of PGL(p 2) , J. Amer. Math. Soc. 23 (2010), 693712.
[Mer13] Merkurjev, A., Essential dimension: A survey , Transform. Groups 18 (2013), 415481.
[Mer16] Merkurjev, A., Essential dimension , in Séminaire Bourbaki, Astérisque, vol. 380 (Société Mathématique de France, 2016), 423448.
[Mer17] 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),∼merkurev/papers/retract-class-space.pdf.
[Pop88] Popov, A. M., Finite isotropy subgroups in general position of simple linear Lie groups , Trans. Moscow Math. Soc. (1988), 205249 (Russian original: Trudy Moskov. Mat. Obschch. 50 (1987), 209–248, 262).
[Pop80] Popov, V. L., Classification of spinors of dimension 14 , Trans. Moscow Math. Soc. 37 (1980), 181232.
[Rei10] Reichstein, Z., Essential dimension , in Proceedings of the international congress of mathematicians 2010 (World Scientific, Singapore, 2010).
[RY00] Reichstein, Z. and Youssin, B., Essential dimensions of algebraic groups and a resolution theorem for G-varieties , Canad. J. Math. 52 (2000), 10181056, with an appendix by J. Kollár and E. Szabó.
[Roh93] Röhrle, G., On certain stabilizers in algebraic groups , Comm. Algebra 21 (1993), 16311644.
[Ros99a] Rost, M., On 14-dimensional quadratic forms, their spinors, and the difference of two octonion algebras, Preprint (1999),∼rost/spin-14.html.
[Ros99b] Rost, M., On the Galois cohomology of Spin(14), Preprint (1999),∼rost/spin-14.html.
[Ste68] Steinberg, R., Lectures on Chevalley groups (Yale University, New Haven, CT, 1968).
[SF88] 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).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO

Spinors and essential dimension

  • Skip Garibaldi (a1) and Robert M. Guralnick (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.