Hostname: page-component-669899f699-qzcqf Total loading time: 0 Render date: 2025-04-26T21:37:15.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper Bounds for the Essential Dimensionof E7

Published online by Cambridge University Press:  20 November 2018

Mark L. MacDonald
Mark L. MacDonald*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Bailrigg, Lancaster LA1 4YF, United Kingdom e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type ${{E}_{7}}$ over a field of characteristic not 2 or 3. In particular, $\text{ed}\left( {{E}_{7}} \right)\,\le \,29$, and $\text{ed}\left( {{E}_{7}};\,2 \right)\,\le \,27$.

Keywords

20G1520G41E7essential dimensionstabilizer in general position
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 795 - 800
Copyright
Copyright © Canadian Mathematical Society 2013

References

[Bo11] Boehning, C., The rationality problem in invariant theory. http://www.math.uni-hamburg.de/home/boehning/research/invarianttheorybook.pdf Google Scholar
[CS06] Chernousov, V. and Serre, J.-P., Lower bounds for essential dimension via orthogonal representations. J. Algebr. 305 (2006), no. 2, 1055–1070. http://dx.doi.org/10.1016/j.jalgebra.2005.10.032 Google Scholar
[Ga01] Garibaldi, S., The Rost invariant has trivial kernel for quasi-split groups of low rank. Comment. Math. Helv. 76 (2001), no. 4, 684–711. http://dx.doi.org/10.1007/s00014-001-8325-8 Google Scholar
[Ga09] Garibaldi, S., Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc. 200 (2009), no. 937.Google Scholar
[GR09] Gille, P. and Reichstein, Z., A lower bound on the essential dimension of a connected linear group. Comment. Math. Helv. 84 (2009), no. 1, 189–212. http://dx.doi.org/10.4171/CMH/158 Google Scholar
[Ig70] Igusa, J., A classification of spinors up to dimension twelve. Amer. J. Math. 92 (1970), 997–1028. http://dx.doi.org/10.2307/2373406 Google Scholar
[Ja68] Jacobson, N., Structure and representations of Jordan algebras. American Mathematical Society Colloquium Publications, XXXIX, American Mathematical Society, Providence, RI, 1968.Google Scholar
[KMRT] Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P., The book of involutions. American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998.Google Scholar
[Le04] Lemire, N., Essential dimension of algebraic groups and integral representations of Weyl groups. Transform. Groups 9 (2004), no. 4, 337–379. http://dx.doi.org/10.1007/s00031-004-9003-x Google Scholar
[Mac11] MacDonald, M. L., Essential p-dimension of the normalizer of a maximal torus. Transform. Groups 16 (2011), no. 4, 1143–1171. http://dx.doi.org/10.1007/s00031-011-9157-2 Google Scholar
[MR09] Meyer, A. and Reichstein, Z., Essential dimension of the normalizer of an maximal torus in the projective linear group. Algebra Number Theory 3 (2009), no. 4, 467–487. http://dx.doi.org/10.2140/ant.2009.3.467 Google Scholar
[Po86] Popov, A. M., Finite isotropy subgroups in general position of simple linear Lie groups. Trans. Moscow Math. Soc. 1986, American Mathematical Society, Providence, RI, 1986, pp. 3–63.Google Scholar
[PV89] Popov, V. L. and Vinberg, E. B., Invariant theory. In: Algebraic Geometry, 4, Encyclopaedia of Mathematical Sciences 55, Springer, 1989.Google Scholar
[Re10] Reichstein, Z., Essential dimension. In: Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188.Google Scholar
[RY00] Reichstein, Z. and Youssin, B., Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Szab ´ o. Canad. J. Math. 52 (2000), no. 5, 1018–1056. http://dx.doi.org/10.4153/CJM-2000-043-5 Google Scholar
[Se02] Serre, J.-P.. Galois cohomology. Translated from the French by Patrick Ion and revised by the author. Corrected reprint of the 1997 English edition. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
[SV00] Springer, T. A. and Veldkamp, F. D., Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.Google Scholar