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A COUNTEREXAMPLE TO A CONJUGACY CONJECTURE OF STEINBERG

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Abstract

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p ≥ 0. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements a, a′ ∈ G are conjugate in G if and only if f(a) and f(a′) are conjugate in GL(V) for every rational irreducible representation f : G → GL(V). Steinberg showed that the conjecture holds if a and a′ are semisimple, and also proved the conjecture when p = 0. In this paper, we give a counterexample to Steinberg’s conjecture. Specifically, we show that when p = 2 and G is simple of type C5, there exist two non-conjugate unipotent elements u, u′ ∈ G such that f(u) and f(u′) are conjugate in GL(V) for every rational irreducible representation f : G→ GL(V).

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References

  1. J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics, Vol. 11, Cambridge University Press, Cambridge, 1986.

  2. P. Bala, R. W. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 401–425.

    Article  MathSciNet  Google Scholar 

  3. P. Bala, R. W. Carter, Classes of unipotent elements in simple algebraic groups. II, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 1, 1–17.

    Article  MathSciNet  Google Scholar 

  4. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265.

    Article  MathSciNet  Google Scholar 

  5. R. Brauer, C. Nesbitt, On the modular characters of groups, Ann. of Math. (2) 42 (1941), 556–590.

  6. A. M. Cohen, S. H. Murray, D. E. Taylor, Computing in groups of Lie type, Math. Comp. 73 (2004), no. 247, 1477–1498.

    Article  MathSciNet  Google Scholar 

  7. W. A. de Graaf, Constructing representations of split semisimple Lie algebras, J. Pure Appl. Algebra 164 (2001), no. 1-2, 87–107.

    Article  MathSciNet  Google Scholar 

  8. W. A. de Graaf, Computation with Linear Algebraic Groups, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2017.

  9. S. Donkin, On tilting modules and invariants for algebraic groups, in: Finite-dimensional Algebras and Related Topics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 59–77.

  10. R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), no. 1, 202–221.

    Article  MathSciNet  Google Scholar 

  11. M. Gerstenhaber, Dominance over the classical groups, Ann. of Math. (2) 74 (1961), 532–569.

  12. W. H. Hesselink, Nilpotency in classical groups over a field of characteristic 2, Math. Z. 166 (1979), no. 2, 165–181.

    Article  MathSciNet  Google Scholar 

  13. D. F. Holt, The Meataxe as a tool in computational group theory, in: The Atlas of Finite Groups: Ten Years on (Birmingham, 1995), London Math. Soc. Lecture Note Ser., Vol. 249, Cambridge Univ. Press, Cambridge, 1998, pp. 74–81.

  14. D. F. Holt, S. Rees, Testing modules for irreducibility, J. Austral. Math. Soc. Ser. A 57 (1994), no. 1, 1–16.

    Article  MathSciNet  Google Scholar 

  15. R. B. Howlett, L. J. Rylands, D. E. Taylor, Matrix generators for exceptional groups of Lie type, J. Symbolic Comput. 31 (2001), no. 4, 429–445.

    Article  MathSciNet  Google Scholar 

  16. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972.

  17. J. E. Humphreys, The Steinberg representation, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 2, 247–263.

    Article  MathSciNet  Google Scholar 

  18. G. Ivanyos, K. Lux, Treating the exceptional cases of the MeatAxe, Experiment. Math. 9 (2000), no. 3, 373–381.

    Article  MathSciNet  Google Scholar 

  19. R. Lawther, Jordan block sizes of unipotent elements in exceptional algebraic groups, Comm. Algebra 23 (1995), no. 11, 4125–4156.

    Article  MathSciNet  Google Scholar 

  20. R. Lawther, Unipotent classes in maximal subgroups of exceptional algebraic groups, J. Algebra 322 (2009), no. 1, 270–293.

    Article  MathSciNet  Google Scholar 

  21. M. W. Liebeck, G. M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, Mathematical Surveys and Monographs, Vol. 180, American Mathematical Society, Providence, RI, 2012.

  22. F. Lübeck, Tables of weight multiplicities, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html, 2017.

  23. G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201–213.

    Article  MathSciNet  Google Scholar 

  24. O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 4, 625–644.

  25. R. A. Parker, The computer calculation of modular characters (the meat-axe), in: Computational Group Theory (Durham, 1982), Academic Press, London, 1984, pp. 267–274.

  26. K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen, J. Algebra 49 (1977), no. 2, 525–536.

    Article  MathSciNet  Google Scholar 

  27. K. Pommerening, Über die unipotenten Klassen reduktiver Gruppen. II, J. Algebra 65 (1980), no. 2, 373–398.

    Article  MathSciNet  Google Scholar 

  28. R. W. Richardson, Jr., Conjugacy classes in Lie algebras and algebraic groups, Ann. of Math. (2) 86 (1967), 1–15.

  29. L. J. Rylands, D. E. Taylor, Matrix generators for the orthogonal groups, J. Symbolic Comput. 25 (1998), no. 3, 351–360.

    Article  MathSciNet  Google Scholar 

  30. G. M. Seitz, The Maximal Subgroups of Classical Algebraic Groups, Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286.

    MathSciNet  MATH  Google Scholar 

  31. I. I. Simion, Centers of Centralizers of Unipotent Elements in Exceptional Algebraic Groups, PhD thesis, EPF Lausanne, 2013.

  32. R. Steinberg, Generators for simple groups, Canad. J. Math. 14 (1962), 277–283.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. (1965), no. 25, 49–80.

  35. R. Steinberg, Classes of elements of semisimple algebraic groups, in: Proc. Internat. Congr. Math. (Moscow, 1966), Mir, Moscow, 1968, pp. 277–284.

  36. R. Steinberg, Conjugacy in semisimple algebraic groups, J. Algebra 55 (1978), no. 2, 348–350.

    Article  MathSciNet  Google Scholar 

  37. R. Steinberg, Lectures on Chevalley Groups, University Lecture Series, Vol. 66, American Mathematical Society, Providence, RI, 2016.

  38. I. D. Suprunenko, The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic, Mem. Amer. Math. Soc. 200 (2009), no. 939, vi+154.

    MathSciNet  MATH  Google Scholar 

  39. D. E. Taylor, Pairs of generators for matrix groups. I, The Cayley Bulletin 3 (1987), 76–85.

    Google Scholar 

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  1. School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK

    MIKKO KORHONEN

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KORHONEN, M. A COUNTEREXAMPLE TO A CONJUGACY CONJECTURE OF STEINBERG. Transformation Groups 25, 1209–1222 (2020). https://doi.org/10.1007/s00031-019-09538-3

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