Abstract
We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isS n , these objects are field extensions; ifG=O n , they are quadratic forms; ifG=PGL n , they are division algebras (all of degreen); ifG=G 2, they are octonion algebras; ifG=F 4, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.
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References
[B] E. Bayer-Fluckiger,Galois cohomology and the trace form, Jahresber. Deutsch. Math.-Verein96 (1994), no. 2, 35–55.
[BP] E. Bayer-Fluckiger, R. Parimala,Galois cohomology of classical groups over fields of cohomological dimension ≤2, Invent. Math.122 (1995), 195–229.
[BK] Ф. Богомолов, П. Кацыло,Рациональносмь некоморых факмор-многообразий, Мат. Сборник126 (168) (1985), No4, 584–589. English translation: F. Bogomolov, P. Katsylo,Rationality of certain quotient varieties, Mat. USSR, Sb.54(1986), 571–576.
[BR1] J. Buhler, Z. Reichstein,On the essential dimension of a finite group, Compositio Math.106 (1997), 159–179.
[BR2] J. Buhler, Z. Reichstein,On Tschirnhaus transformations, in “Number Theory”, Proceedings of a conference held at Penn. State University, edited by S. Ahlgren, G. Andrews and K. Ono, Kluwer Acad. Publishers, 127–142, 1999. Preprint available at http://www.orst.edu/~reichstz/pub.html.
[De] A. Delzant,Définition des classes de Stiefel-Whitney d'un module quadratique sur un corps de caractéristique différente de 2, C. R. Acad. Sci. Paris255 (1962), 1366–1368.
[Do] I. V. Dolgachev,Rationality of fields of invariants, in Algebraic Geometry, Bowdoin 1985, Proc. of Symp. in Pure Math., vol. 46, part 2, AMS, 1987, 3–16.
[Gi] Ph. Gille,Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2, Compositio Math., to appear.
[G] A. Grothendieck,Torsion homologique et sections rationnelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP, 1958.
[H] J. E. Humphreys,Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, 1975. Russian translation: Дж. Хамфри,Линейные алгебраические группы, М., Наука, 1980.
[I] A. V. Iltyakov,On rational invariants of the group E 6, Proc. Amer. Math. Soc.124 (1966), no. 12, 3637–3640.
[J] N. Jacobson,Structure and Representations of Jordan Algebras, AMS Colloq. Publ.39, Amer. Math. Soc., Providence, Rhode Island, 1968.
[Ka1] П. И. Кацыло,Рациональносмь просмрансмв орбим неприводимых предсмаблений группы SL2, Изв. АН СССР, сер. мат.47 (1983), 26–36. English translation: P. I. Katsylo,Rationality of the orbit spaces of irreducible representations of the group SL2, Math. USSR-Izvestya22 (1984), no. 1, 23–32.
[Ka2] P. I. Katsylo,Rationality of the module variety of mathematical instantons with c 2=5, in Lie Groups, their Discrete Subgroups, and Invariant Theory, Advances in Soviet Math.8 (1992), Amer. Math. Soc., Providence, RI., 65–68.
[KMRT] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, AMS Colloquium Publications, vol. 44, 1998.
[Ko1] В. Е. Кордонский,Сущесмвенная размерносмь алгебраических групп, preprint, 1999.
[Ko2] В. Е. Кордонский,О сущесмвенной размерносми и гипомезе Серра II для исключимельных групп, preprint, 1999.
[Pf] A. Pfister,Quadratic Forms with Applications to Geometry and Topology, Cambridge University Press, 1995.
[Pi] R. S. Pierce,Associative Algebras, Springer, New York, 1982. Russian translation: С. Пирс,Ассоциамивные алгебры, М., Мир, 1986.
[Po] V. L. Popov,Sections in invariant theory, Proceedings of the Sophus Lie Memorial Conference, Scandinavian University Press, 1994, 315–362.
[PV] Э. Б. Винберг, В. Л. Попов,Теория инварианмов, Соврем. проблемы математики. Фунд. направл., ВИНИТИ, Москва, т. 55, 1989, 137–309. English translation: V. L. Popov, E. B. Vinberg,Invariant Theory, Algebraic Geometry IV, Encyclopedia of Math. Sciences, Vol. 55, Springer-Verlag, 1994, pp. 123–284
[Pr1] C. Procesi,Non-commutative affine rings, Atti Acc. Naz. Lincei, S. VIII, v. VIII, fo. 6 (1967), 239–255.
[Pr2] C. Procesi,The invariant theory of n × n-matrices, Adv. Math.19 (1976), 306–381.
[Re] Z. Reichstein,On a theorem of Hermite and Joubert, Canad. J. Math.51(1) (1999), 69–95.
[RY1] Z. Reichstein, B. Youssin,Essential dimensions of algebraic groups and a resolution theorem for G-varieties, with an appendix by J. Kollár and E. Szabó. Canadian J. Math.52(5) (2000). Preprint available at http://www.orst.edu/~reichstz/pub.html.
[RY2] Z. Reichstein, B. Youssin,On a property of special groups, MSRI preprint 2000-010, available at http://msri.org/publications/preprints/online/2000-010.html.
[Ros1] M. Rosenlicht,Some basic theorems on algebraic groups, Amer. J. of Math.78 (1956), 401–443.
[Ros2] M. Rosenlicht,A remark on quotient spaces, Anais da Academia Brasileira de Ciências35 (1963), 487–489.
[Rost1] M. Rost,A (mod 3)invariant of exceptioal Jordan algebras, C. R. Acad. Sci. Paris Sér. I Math.313 (1991), no. 12, 823–827.
[Rost2] M. Rost,On Galois cohomology of Spin(14), preprint, March 1999. Available at http://www.physik.uni-regensburg.de/~rom03516.
[Row] L. H. Rowen,Brauer factor sets and simple algebras, Trans. Amer. Math. Soc.282 (1984), no. 2, 765–772.
[Sa] D. J. Saltman,Lectures on Division Algebras, CBMS Regional Conferences Series in Mathematics94, Amer. Math. Soc., 1999.
[Sc] R. D. Schaefer, An Introduction to Nonassociative Algebras, Academic Press, New York and London, 1966.
[Se1] J.-P. Serre,Espaces fibrés algébriques, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP, 1958.
[Se2] J.-P. Serre,Local Fields, Springer-Verlag, 1979.
[Se3] J.-P. Serre,Cohomologie galoisienne: progrès et problèmes, in “Séminaire Bourbaki, Volume 1993/94, Exposés 775-789”, Astérisque227 (1995), 229–257.
[Se4] J.-P. Serre,Galois Cohomology, Springer, 1997.
[SSSZ] К. А. Жевлаков, А. М. Слинько, И. П. Шестаков, А. И. Ширшов,Кольца, близкие к ассоциамивным, М., Наука, 1978. English transaltion: I. P. Shestakov, A. I. Shirshov, A. M. Slin'ko, K. A. Zhevlakov, Rings that are Nearly Associative, Academic Press, 1982.
[St] R. Steinberg,Regular elements of semisimple groups, Publ. Math. I.H.E.S.25 (1965), 281–312. Reprinted in [Se4], pp. 155–186.
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Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675
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Reichstein, Z. On the notion of essential dimension for algebraic groups. Transformation Groups 5, 265–304 (2000). https://doi.org/10.1007/BF01679716
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DOI: https://doi.org/10.1007/BF01679716