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GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS

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Abstract

We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero. This relies on bounds on dim V obtained in prior work (part I), which reduce the problem to a finite number of possibilities for G and highest weights for V , but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position.

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References

  1. Е. М. Андреев, В. Л. Попов, О стационарных подгруппах токвея обвего поло в пространстве представлени полупросто группы Ли, Φyʜкц. анализ и его прил. 5 (1971), вып.. 4, 1–8. Engl. transl.: E. M. Andreev, V. L. Popov, Stationary subgroups of points of general position in the representation space of a semisimple Lie group, Functional Anal. Appl. 5 (1971), no. 4, 265–271.

  2. Е. М. Андреев. Э. Винберг, А. Г. Лакшвили, Орбиты наибольwей pазмерности полупростых линеийых групп Ли, Φyʜкц. анализ и его прил. 1 (1967), вып. 4, 3–7. Engl. transl.: E. M. Andreev, E. B. Vinberg, A. G. Elashvili, Orbits of greatest dimension in semi-simple linear Lie groups, Funct. Anal. Appl. 1 (1968), 257–261.

  3. A. Auld, Exceptional modular representations of special linear Lie algebras, PhD thesis, University of Manchester, February 2001.

  4. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: the user language, J. Symbolic Computation 9 (1990), 677–698.

    Article  MathSciNet  Google Scholar 

  5. P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 (2010), no. 1, 533–544.

  6. T. Burness, R. Guralnick, J. Saxl, On base sizes for algebraic groups, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2269–2341.

    Article  MathSciNet  Google Scholar 

  7. P.-E. Chaput, M. Romagny, On the adjoint quotient of Chevalley groups over arbitrary base schemes, J. Inst. Math. Jussieu 9 (2010), no. 4, 673–704.

    Article  MathSciNet  Google Scholar 

  8. B. Conrad, Reductive group schemes, Autour des Schémas en Groupes I, Panoramas et synthèses, vol. 42–43, Sociétè Mathématique de France, 2014.

  9. B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive Groups, 2nd ed., Cambridge University Press, 2015.

  10. M. Demazure, P. Gabriel, Groupes Algébriques. Tome I: Géométrie Algéb-rique, Généralités, Groupes Commutatifs, Masson, Paris, 1970.

  11. M. Demazure, A. Grothendieck, Schémas en groupes II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, Vol. 152, Springer, Berlin, 1970.

  12. А. Г. Элaкшвили, Кanonuecкuй ɕud и стационарные подалгебры токек oбweso nolo eni для простых линий групп Ли, Φyʜкц. анализ и его прил. 6 (1972), вып. 1, 51–62. Engl. transl.: A. G. Èlašvili, Canonical form and stationary subalgebras of points in general position for simple linear Lie groups, Funct. Anal. Appl. 6 (1972), no. 1, 44–53.

  13. S. Garibaldi, R. M. Guralnick, Simple groups stabilizing polynomials, Forum of Mathematics: Pi 3 (2015), e3 (41 pages).

    Article  MathSciNet  Google Scholar 

  14. S. Garibaldi, R. M. Guralnick, Spinors and essential dimension, Compositio Math. 153 (2017), 535–556, with an appendix by A. Premet.

  15. S. Garibaldi, R. M. Guralnick, Generically free representations I: Large representations, to appear in Algebra & Number Theory, arxiv:1711.05502v3 (2017).

  16. S. Garibaldi, R. M. Guralnick, Generically free representations III: Extremely bad characteristic, to appear in Transformation Groups.

  17. S. Garibaldi, R. Parimala, J.-P. Tignol, Discriminant of symplectic involutions, Pure Applied Math. Quarterly 5 (2009), no. 1, 349–374.

    Article  MathSciNet  Google Scholar 

  18. R. Gow, A. Kleshchev, Connections between representations of the symmetric group and the symplectic group in characteristic 2, J. Algebra 221 (1999), 60–89.

    Article  MathSciNet  Google Scholar 

  19. M. Guerreiro, Exceptional representations of simple algebraic groups in prime characteristic, PhD thesis, University of Manchester, 1997, arxiv:1210.6919 (2012).

  20. R. M. Guralnick, R. Lawther, Generic stabilizers in actions of simple algebraic groups I: modules and first Grassmannian varieties, arxiv:1904.13375v2 (2019).

  21. R. M. Guralnick, M. W. Liebeck, D. Macpherson, G.M. Seitz, Modules for algebraic groups with finitely many orbits on subspaces, J. Algebra 196 (1997), 211–250.

    Article  MathSciNet  Google Scholar 

  22. J. E. Humphreys, Algebraic Groups and Modular Lie Algebras, Memoirs of the American Mathematical Society, No. 71, American Mathematical Society, Providence, R.I., 1967.

  23. J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.

  24. N. Karpenko, Canonical dimension, Proceedings of the International Congress of Mathematicians 2010, World Scientific, 2010.

  25. P. Levy, Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices, Transform. Groups 14 (2009), no. 2, 417–461.

    Article  MathSciNet  Google Scholar 

  26. M. Liebeck, G. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, Mathematical Surveys and Monographs, no. 180, American Mathematical Society, Providence, RI, 2012.

  27. R. Lötscher, A fiber dimension theorem for essential and canonical dimension, Compositio Math. 149 (2013), no. 1, 148–174.

    Article  MathSciNet  Google Scholar 

  28. R. Lötscher, M. MacDonald, A. Meyer, Z. Reichstein, Essential dimension of algebraic tori, J. Reine Angew. Math. 677 (2013), 1–13.

    Article  MathSciNet  Google Scholar 

  29. F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169.

    Article  MathSciNet  Google Scholar 

  30. G. J. McNinch, Semisimplicity of exterior powers of semisimple representations of groups, J. Algebra 225 (2000), 646–666.

    Article  MathSciNet  Google Scholar 

  31. A. Merkurjev, Essential dimension: a survey, Transform. Groups 18 (2013), no. 2, 415–481.

    Article  MathSciNet  Google Scholar 

  32. А. М. Попов, Конечные стационарные подгруппы oбweso полоcenu яnростых линийных групп Ли, Тр. MMO 48 (1985), 7–59. Engl. transl.: A. M. Popov, Finite isotropy subgroups in general position of simple linear Lie groups, Trans. Moscow Math. Soc. 1986 (1988), 3–63.

  33. Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Алгебраицкая гео-мeтpия–4, Итоги науки и техн., сер. Совр. пробл. мат., Φyнnдaм.направл., т. 55, ВИНИТИ, М., 1989, 137–309. Engl. transl.: V. L. Popov, E. B. Vinberg, Invariant theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Algebraic Geometry IV, Springer-Verlag, Berlin, 1994, pp. 123–284.

  34. A. A. Premet, I. D. Suprunenko, The Weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights, Comm. Algebra 11 (1983), no. 12, 1309–1342.

    Article  MathSciNet  Google Scholar 

  35. M. Reeder, P. Levy, J.-K. Yu, B. H. Gross, Gradings of positive rank on simple Lie algebras, Transform. Groups 17 (2012), no. 4, 1123–1190.

    Article  MathSciNet  Google Scholar 

  36. Z. Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians 2010, World Scientific, 2010.

  37. R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56.

    Article  MathSciNet  Google Scholar 

  38. R. Steinberg, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc. Vol. 80, 1968.

  39. H. Strade, R. Farnsteiner, Modular Lie Algebras and Their Representations, Monographs and Textbooks in Pure and Applied Math., Vol. 116, Marcel Dekker, New York, 1988.

  40. Э. Б. Винберг, Группа Вейя градуированной алгебры Ли, Изв. СССР. Cep. матем. 40 (1976), вып. 3, 488–526. Engl. transl.: E. B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR Izv. 10 (1976), 463–495.

  41. Э. Б. Винберг, А. Г. Эдаксвили, Классификаци тривекторов девитай мерного Пространства, Труды семинара по вект. и тенз. анализy 18 (1978), 197–233. Engl. transl.: E. B. Vinberg, A. G. Elashvili, Classification of trivectors of a 9-dimensional space, Sel. Math. Sov. 7 (1988), no. 1, 63–98.

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Authors and Affiliations

  1. IDA Center for Communications Research-La Jolla, 4320 Westerra Ct, San Diego, CA, 92121, USA

    SKIP GARIBALDI

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    ROBERT M. GURALNICK

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Partially supported by NSF grants DMS-1600056 and DMS-1901595.

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GARIBALDI, S., GURALNICK, R.M. GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS. Transformation Groups 25, 793–817 (2020). https://doi.org/10.1007/s00031-020-09591-3

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