Startseite Mathematik Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes
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Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes

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Veröffentlicht/Copyright: 8. August 2017

Abstract

We investigate the saturation rank of a finite group scheme defined over an algebraically closed field 𝕜 of positive characteristic p. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group SLn.

MSC 2010: 17B50; 14L15

Acknowledgements

The results of this paper are part of the author’s doctoral thesis written at the University of Kiel. He would like to thank his advisor, Rolf Farnsteiner, for his continuous support. Furthermore, he thanks the members of his working group for proofreading the paper and the referee for useful comments.

References

[1] D. J. Benson, Representations and Cohomology. II: Cohomology of Groups and Modules, Cambridge Stud. Adv. Math. 31, Cambridge University Press, Cambridge, 1991. Suche in Google Scholar

[2] J. F. Carlson, E. M. Friedlander and J. Pevtsova, Elementary subalgebras of Lie algebras, J. Algebra 442 (2015), 155–189. 10.1016/j.jalgebra.2014.10.015Suche in Google Scholar

[3] J. F. Carlson, Z. Lin, D. K. Nakano and B. J. Parshall, The restricted nullcone, Combinatorial and Geometric Representation Theory (Seoul 2001), Contemp. Math. 325, American Mathematical Society, Providence (2003), 51–75. 10.1090/conm/325/05664Suche in Google Scholar

[4] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Math. Ser., Van Nostrand Reinhold, New York, 1993. Suche in Google Scholar

[5] M. Demazure and P. Gabriel, Groupes Algébriques, Masson, Paris, 1970. Suche in Google Scholar

[6] L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239. 10.1090/S0002-9947-1961-0137742-1Suche in Google Scholar

[7] R. Farnsteiner, Varieties of tori and Cartan subalgebras of restricted Lie algebras, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4181–4236. 10.1090/S0002-9947-04-03476-2Suche in Google Scholar

[8] R. Farnsteiner, Combinatorial and geometric aspects of the representation theory of finite group schemes, Lie Theory and Representation Theory, Surv. Mod. Math. 2, International Press, Somerville (2012), 47–149. Suche in Google Scholar

[9] R. Farnsteiner, Jordan types for indecomposable modules of finite group schemes, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 925–989. 10.4171/JEMS/452Suche in Google Scholar

[10] R. Farnsteiner, Representations of finite group schemes and morphisms of projective varieties, Proc. Lond. Math. Soc. (3) 114 (2017), no. 3, 433–475. 10.1112/plms.12010Suche in Google Scholar

[11] E. M. Friedlander and J. Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005), no. 2, 379–420. 10.1353/ajm.2005.0010Suche in Google Scholar

[12] E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. 10.1007/s002220050119Suche in Google Scholar

[13] I. Gordon and A. Premet, Block representation type of reduced enveloping algebras, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1549–1581. 10.1090/S0002-9947-01-02826-4Suche in Google Scholar

[14] J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Math. Surveys Monogr. 43, American Mathematical Society, Providence, 1995. Suche in Google Scholar

[15] J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Math. Surveys Monogr. 107, American Mathematical Society, Providence, 2003. Suche in Google Scholar

[16] J. C. Jantzen, Nilpotent orbits in representation theory, Lie Theory, Progr. Math. 228, BirkhĂ€user, Boston (2004), 1–211. 10.1007/978-0-8176-8192-0_1Suche in Google Scholar

[17] E. Letellier, Fourier Transforms of Invariant Functions on Finite Reductive LIe Algebras, Lecture Notes in Math. 1859, Springer, Berlin, 2005. 10.1007/b104209Suche in Google Scholar

[18] P. Levy, Commuting varieties of Lie algebras over fields of prime characteristic, J. Algebra 250 (2002), no. 2, 473–484. 10.1006/jabr.2001.9083Suche in Google Scholar

[19] G. J. McNinch, Optimal SLⁱ(2)-homomorphisms, Comment. Math. Helv. 80 (2005), no. 2, 391–426. 10.4171/CMH/19Suche in Google Scholar

[20] G. J. McNinch and D. M. Testerman, Nilpotent centralizers and Springer isomorphisms, J. Pure Appl. Algebra 213 (2009), no. 7, 1346–1363. 10.1016/j.jpaa.2008.12.007Suche in Google Scholar

[21] A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), no. 3, 653–683. 10.1007/s00222-003-0315-6Suche in Google Scholar

[22] D. Quillen, The spectrum of an equivariant cohomology ring. I, Ann. of Math. (2) 94 (1971), 549–572. 10.2307/1970770Suche in Google Scholar

[23] D. Quillen, The spectrum of an equivariant cohomology ring. II, Ann. of Math. (2) 94 (1971), 573–602 10.2307/1970771Suche in Google Scholar

[24] A. Suslin, E. M. Friedlander and C. P. Bendel, Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), no. 3, 693–728. 10.1090/S0894-0347-97-00240-3Suche in Google Scholar

[25] A. Suslin, E. M. Friedlander and C. P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729–759. 10.1090/S0894-0347-97-00239-7Suche in Google Scholar

[26] B. B. Venkov, Cohomology algebras for some classifying spaces (in Russian), Dokl. Akad. Nauk. SSSR 127 (1959), 943–944. Suche in Google Scholar

[27] B. B. Venkov, Characteristic classes for finite groups (in Russian), Dokl. Akad. Nauk. SSSR 137 (1961), 1274–1277. Suche in Google Scholar

[28] W. C. Waterhouse, Introduction to Affine Group Schemes, Grad. Texts in Math. 66, Springer, New York, 1979. 10.1007/978-1-4612-6217-6Suche in Google Scholar

Received: 2017-1-11
Revised: 2017-7-22
Published Online: 2017-8-8
Published in Print: 2018-3-1

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Heruntergeladen am 4.2.2026 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0007/pdf
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