Computational Aspects of the Isomorphism Problem

  • Frauke M. Bleher
  • Wolfgang Kimmerle
  • Klaus W. Roggenkamp
  • Martin Wursthor
Conference paper

Abstract

The aim of this article is to lead the reader on a journey through the representation theory of finite groups of Lie type and Hecke algebras. We will present some basic results obtained in recent years, explain the ideas behind them, and give lots of examples; proofs are usually omitted but we provide explicit references to an extensive bibliography.

Keywords

Ghost 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Alvis andG. Lusztig,The representations and generic degrees of the Hecke algebra of type H4 J. reine angew. Math. 336 (1982), 201–212; Correction: ibid. 449 (1994), 217-218.Google Scholar
  2. 2.
    H.H. Anderson, J.C. Jantzen, and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Asterisque 220 (1884).Google Scholar
  3. 3.
    S. Ariki, On the decomposition numbers of the Hecke algebra of G(m, 1, n), J. of Math. Kyoto Univ. 36 (1996), 789–808.Google Scholar
  4. 4.
    C.T. Benson and C.W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273; Corrections and additions, ibid. 202 (1975), 405-406.Google Scholar
  5. 5.
    K. Bremke, The decomposition numbers of Hecke algebras of type F4 with unequal parameters, Manuscripta Math. 83 (1994), 331–346.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    M. Broue, Isometries parfaites, types de blocs, categories derivees, in Representations Lineaires de Groupes Finis, Asterisque 181-182 (1990), 61–92.MathSciNetGoogle Scholar
  7. 7.
    M. Broue and G. Malle, Theoremes de Sylow generiques pour les groupes reductifs sur les corps finis,Math. Ann. 292 (1992), 241–262.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M. Broue and G. Malle, Zyklotomische Heckealgebren, in: Representations unipotentes generiques et blocs des groupes reductifs finis, Asterisque 212 (1993), 119–189.MathSciNetGoogle Scholar
  9. 9.
    M. Broue, G. Malle, and J. Michel, Generic blocks of finite reductive groups, in: Representations unipotentes generiques et blocs des groupes reductifs finis Asterisque 212 (1993), 7–92.MathSciNetGoogle Scholar
  10. 10.
    M. Broue and J. Michel, Blocs et series de Lusztig dans un groupe reductive fini,J. reine angew. Math. 395 (1989), 56–67.MathSciNetMATHGoogle Scholar
  11. 11.
    M. Broue and J. Michel, Blocs it groupes de de£aut abeliens des groupes reductifs finis, in: Representations unipotentes generiques et blocs des groupes reductifs finis, Asterisque 212 (1993), 93–118.MathSciNetGoogle Scholar
  12. 12.
    M. Cabanes, A criterion for complete reducibility and some applications, in: Representations lineaires de groupes finis, Luminy, 16-21 mai 1988, Asterisque 181-182 (1990), 93–112.MathSciNetGoogle Scholar
  13. 13.
    M. Cabanes and M. Enguehard, Local methods for blocks ofreductive groups over a finite field, in: Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 141–163. Birkhauser, Basel, 1997.Google Scholar
  14. 14.
    R.W. Carter, Representation theory of the O-Heeke algebra, J. Algebra 104 (1986), 89–103.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    R.W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, Chichester, 1985.MATHGoogle Scholar
  16. 16.
    C. W. Curtis and I. Reiner, Methods of representation theory, vols. 1,2 Wiley, Chichester, 1981, 1987.Google Scholar
  17. 17.
    P. Deligne and G. Lusztig, Duality for representations of a reductive group over a finite field II, J. Algebra 81 (1983), 540–545.Google Scholar
  18. 18.
    R. Dipper, On the decomposition numbers of the finite general linear groups, Trans. Amer. Math. Soc. 290 (1985), 315–343; II, 292 (1985), 123-133.Google Scholar
  19. 19.
    R. Dipper, On quotients of Hom-functors and representations of finite general linear groups I, J. Algebra 130 (1990), 235–259.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    R. Dipper, On quotients of Hom-functors and representations of finite general linear groups II, Preprint, Stuttgart 1997.Google Scholar
  21. 21.
    R. Dipper and S. Donkin, Quantum GLn Proc. London Math. Soc. 63 (1991), 165–211.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    R. Dipper and J. Du, Harish-Chandra vertices,J. reine angew. Math. 437 (1993), 101–130.MathSciNetMATHGoogle Scholar
  23. 23.
    R. Dipper and J. Du, Harish-Chandra vertices and Steinberg’s tensor product theorem for general linear groups, Proc. London Math. Soc. 75 (1997), 559–599.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    R. Dipper and P. Fleischmann, Modular Harish-Chandra theory I, Math. Z. 211 (1992), 49–71; II, Arch. Math. 62 (1994), 26–32.Google Scholar
  25. 25.
    R. Dipper and J. Gruber, Generalized q-Schur algebras and modular representation theory of finite groups with split (B, N)-pairs, Preprint, Stuttgart 1997.Google Scholar
  26. 26.
    R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    R. Dipper and G. D. James, Identification of the irreducible modular representations of GLn(q), J. Algebra 104 (1986), 266–288.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    R. Dipper and G. D. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. 54 (1989), 57–82.MathSciNetCrossRefGoogle Scholar
  29. 29.
    R. Dipper and G. D. James, The q-Schur algebra, Proc. London Math. Soc. 59 (1989), 23–50.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    R. Dipper and G. D. James, q-tensor space and q-Weyl modules, Proc. Amer. Math. Soc. 327 (1991), 251–282.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    R. Dipper and G.D. James, Representations of Heeke algebras of type Bn, J. Algebra 146 (1992), 454–48l.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    R.Dipper, G.D.James, andA.Mathas, The (Q,q)-Schur algebra, Proc. London Math. Soc., to appear.Google Scholar
  33. 33.
    R. Dipper, G.D. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., to appear.Google Scholar
  34. 34.
    R. Dipper, G.D. James, and G.E. Murphy, Heeke algebras of type Bnat roots of unity,Proc. London Math. Soc. 70 (1995), 505–528.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    J. Du, Kazhdan-Lusztig bases and isomorphism theorems for q-Schur algebras, Contemp. Math. 139 (1992), 121–140.Google Scholar
  36. 36.
    J. Du, A new proof for the canonical bases of type A, Preprint, 1996.Google Scholar
  37. 37.
    J. Du, B. Parshall, and L. Scott, Stratifying endomorphism algebras associated to Heeke algebras, Preprint, Sydney 1996.Google Scholar
  38. 38.
    J.DuandH.Rui, Based algebras and standard basis for quasi-hereditary algebras, Trans. Amer. Math. Soc., to appear.Google Scholar
  39. 39.
    J. Du and L. Scott, The q-Schur2 algebra, Preprint, Charlottesville 1997.Google Scholar
  40. 40.
    P. Fong and B. Srinivasan, Generalized Harish-Chandra theory for unipotent characters of finite classical groups,J. Algebra 104(1986), 301–309.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    P. Fong and B. Srinivasan, The blocks of finite classical groups,J. reine angew. Math. 396 (1989), 122–191.MathSciNetMATHGoogle Scholar
  42. 42.
    P. Fong and B. Srinivasan, Brauer trees in classical groups,J. Algebra 131(1990), 179–225.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    M. Geck, Irreducible Brauer characters of the 3-dimensional special unitary groups in non-defining characteristic, Comm. Algebra 18(1990), 563–584.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    M. Geck, On the decomposition numbers of the finite unitary groups in non-defining characteristic, Math. Z. 207 (1991), 83–89.MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    M. Geck, Generalized Gelfand-Graev characters for Steinberg’s triality groups and their applications, Comm. Algebra 19(1991), 3249–3269.MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    M. Geck, Brauer trees of Hecke algebras, Comm. Algebra 20(1992), 2937–2973.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    M. Geck, The decomposition numbers of the Hecke algebra of type E6, Math. Comp. 61 (1993), 889–899.MathSciNetMATHGoogle Scholar
  48. 48.
    M. Geck, Basic sets of Brauer characters of finite groups of Lie type, II,J. London Math. Soc. 47 (1993), 255–268; III, Manuscripta Math. 85 (1994), 195-216.Google Scholar
  49. 49.
    M. Geck, On the character values of Iwahori-Hecke algebras of exceptional type, Proc. London Math. Soc. 68 (1994), 51–76.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    M. GECK, Beitriige zur Darstellungstheorie von Iwahori-Hecke-Algebren, Habilitationsschrift, Aachen 1994, Aachener Beitrage zur Mathematik, Band 11, Aachen, 1995.Google Scholar
  51. 51.
    M. Geck, Kazhdan-Lusztig cells and decomposition numbers, Institut de Mathématiques de Jussieu, Paris, Prépubl. 147, 1997.Google Scholar
  52. 52.
    M. Geck and G. Hiss, Basic sets of Brauer characters of finite groups of Lie type,J. reine angew. Math. 418 (1991),173–188.MathSciNetMATHGoogle Scholar
  53. 53.
    M. Geck and G. Hiss,Modular representations of finite groups of Lie type in non-defining characteristic, in Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 195–249. Birkhauser, Basel, 1997.Google Scholar
  54. 54.
    M. Geck, G. Hiss, F. Lubeck, G. Malle, and G. Pfeiffer, CHEVIE - A system for computing and processing generic character tables, AAECC 7 (1996), 175–210.MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    M. Geck, G. Hiss, and G. Malle, Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), 182–220.MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    M. Geck, G. Hiss, and G. Malle, Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353–386.MathSciNetMATHGoogle Scholar
  57. 57.
    M. Geck and K. Lux, The decomposition numbers of the Hecke algebra of type F4 Manuscripta Math. 70 (1991), 285–306.MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    M. Geck and G. Malle, Cuspidal unipotent classes and cuspidal Brauer characters, J. London Math. Soc. 53 (1996), 63–78.MathSciNetMATHGoogle Scholar
  59. 59.
    M. Geck and J. Michel, Good elements in finite Coxeter groups and representations of Iwahori-Hecke algebras,Proc. London Math. Soc. (3)(1997), 275–305.MathSciNetCrossRefGoogle Scholar
  60. 60.
    M. Geck and G. Pfeiffer, On the irreducible characters of Heeke algebras,Adv. in Math. 102 (1993), 79–94.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, in Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 251–272. Birkhaser, Basel, 1997.Google Scholar
  62. 62.
    J.J. Graham and G.I. Lehrer, Cellular algebras,Invent. Math. 123 (1996), 1–34.MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    R.M. Green, q-Schur algebras and quantized enveloping algebras, Thesis, University of Warwick, 1995.Google Scholar
  64. 64.
    R.M. Green, Hyperoctahedral Schur algebras, J Algebra 192(1997), 418–438.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    R.M. Green, The affine q-Schur algebra, Preprint, Oxford, 1997.Google Scholar
  66. 66.
    I. Grojnowski, Representations of affine Hecke algebras (and quantum GLn) at roots of unity, Internat. Math. Research Notices 5(1994), 215–217.MathSciNetCrossRefGoogle Scholar
  67. 67.
    J. Gruber, Green vertex theory, Green correspondence and Harish-Chandra induction,J. Algebra 186(1997), 476–521.MathSciNetCrossRefGoogle Scholar
  68. 68.
    J. Gruber and G. Hiss, Decomposition numbers of finite classical groups for linear primes, J. reine angew. Math. 485 (1997), 55–91.MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    T. Halverson and A. Ram, Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type, Trans. Amer. Math. Soc. 348 (1996), 3967–3995.MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    HARISH-CHANDRA, Eisenstein series over finite fields, in Functional analysis and related fields, pp. 76–88. Springer, New York 1970.Google Scholar
  71. 71.
    G. Hiss, On the decomposition numbers of G2(q), J. Algebra 120 (1989), 339–360.MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, RWTH Aachen, 1990.Google Scholar
  73. 73.
    G. Hiss, The Brauer trees of the Ree groups,Comm. Algebra 19 (1991),871–888.MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    G. Hiss, Harish-Chandra series of Brauer characters in a finite group with a split BN-pair, J. London Math. Soc. 48 (1993), 219–228.MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    G. Hiss, Supercuspidal representations of finite reductive groups, J. Algebra 184 (1996), 839–851.MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    G. Hiss and F. Lubeck, The Brauer trees of the exceptional Chevalley groups of types F4and E6 Arch. Math. 70 (1998), 16–21.MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    G. Hiss, F. Lübeck, and G. Malle, The Brauer trees of the exceptional Chevalley groups of type E6, Manuscripta Math. 87 (1995), 131–144.MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    G. Hiss and J. Shamash, 3-Blocks and 3-decomposition numbers of G2 (q), J. Algebra 131(1990), 371–387.MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    G. Hiss and J. Shamash, 2-Blocks and 2-decomposition numbers of the Chevalley groups G2(q), Math. Compo 59(1992), 645–672.MathSciNetMATHGoogle Scholar
  80. 80.
    R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalized Heeke rings, Invent. Math. 58 (1980), 37–64.MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    R. B. Howlett and G. I. Lehrer, On Harish-Chandra induction for modules of Levi subgroups, J. Algebra 165 (1994), 172–183.MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    N. Iwahori, On the structure of the Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 10 (1964), 215–236.MathSciNetMATHGoogle Scholar
  83. 83.
    G.D. James, Unipotent representations of the finite general linear groups, J. Algebra 74(1982), 443–465.MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    G.D. James, Representations of general linear groups, Cambridge Univ. Press, Cambridge, 1984.MATHCrossRefGoogle Scholar
  85. 85.
    G.D. James, The irreducible representations of the finite general linear groups, Proc. London Math. Soc. 52 (1986), 236–268.MATHCrossRefGoogle Scholar
  86. 86.
    G.D. James, The decomposition matrices of GLn(q) for n ~ 10, Proc. London Math. Soc. 60 (1990), 225–265.MATHCrossRefGoogle Scholar
  87. 87.
    G.D. James and A. Mathas, A q-analogue of the Jantzen-Schaper theorem,Proc. London Math. Soc. 74 (1997), 241–274.MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras,Invent. Math. 40 (1979), 165–184.MathSciNetCrossRefGoogle Scholar
  89. 89.
    P. Landrock and G. O. Michler, Principal 2-bloeks of the simple groups of Ree type, Trans. Amer. Math. Soc. 260 (1980), 83–111.MathSciNetMATHGoogle Scholar
  90. 90.
    A. Lascoux, B. Leclerc, and J.-Y. Thibon, Une conjecture pour Ie calcul des matrices de decomposition des algebres de Hecke du type A aux racines del’unite,C. R. Acad. Sci. Paris Ser. 1321 (1995), 511–516.MathSciNetGoogle Scholar
  91. 91.
    A. Lascoux, B. Leclerc, and J.-Y. Thibon, Heeke algebras at roots of unity and crystal bases of quantum affine algebras,Comm. Math. Physics 181(1996), 205–263.MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    B. Leclerc and J.-Y. Thibon, Canonical bases of q-deformed Foek spaces, Internat. Math. Research Notices 9(1996), 447–456.MathSciNetCrossRefGoogle Scholar
  93. 93.
    S.A. Linton, On vector enumeration, Linear AIgebm Appl. 192 (1993), 235–248.MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    G. Lusztig, On a theorem of Benson and Curtis, J. AIgebm 71(1981), 490–498.MathSciNetMATHGoogle Scholar
  95. 95.
    G. Lusztig, Chamcters o/reductive groups over a finite field, Ann. Math. Studies 107,Princeton U. Press, Princeton, 1984.Google Scholar
  96. 96.
    G. Lusztig, On the representations of reductive groups with disconnected centre, Asterisque 168(1988), 157–166.MathSciNetGoogle Scholar
  97. 97.
    G. Lusztig and N. Spaltenstein, Induced unipotent classes,J. London Math. Soc. 19 (1979), 41–52.MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    R. P. Martineau, On representations of the Suzuki groups over fields of odd characteristics, J. London Math. Soc. 6 (1972), 153–160.MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    A. Mathas, Canonical bases and the decomposition numbers of Ariki-Koike algebras, Preprint, London, 1996.Google Scholar
  100. 100.
    J. Muller, Zerlegungszahlen for generische Iwahori-Hecke-Algebren von exzeptionellem Typ, Dissertation, RWTH Aachen, 1995.Google Scholar
  101. 101.
    J. Muller, Decomposition numbers for generic Iwahori-Heeke algebras of non crystallographic type, J. AIgebm 189(1997), 125–149.Google Scholar
  102. 102.
    G.E. Murphy, On the representation theory of the symmetric groups and associated Heeke algebras, J. Algebm 152 (1992) 492–513.MATHCrossRefGoogle Scholar
  103. 103.
    G.E. Murphy, The representations of Heeke algebras oftype An, J. AIgebm 173 (1995), 97–121.MATHGoogle Scholar
  104. 104.
    S. Oehms, Symplektische q-Schur-Algebren, Dissertation, Stuttgart, 1997.MATHGoogle Scholar
  105. 105.
    T. Okuyama and K. Waki, Decomposition numbers of Sp(4,q), J. AIgebm 199 (1998), 544–555.MathSciNetMATHGoogle Scholar
  106. 106.
    C. Pallikaros, Representations of Heeke algebras of type Dn , J. AIgebm 169 (1994), 20–48.MathSciNetMATHGoogle Scholar
  107. 107.
    R. A. Parker, The computer calculation of modular characters (the Meat-Axe), in Computational group theory (ed. M.D. Atkinson). Academic Press, London, 1984.Google Scholar
  108. 108.
    B. Parshall and J.P. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439 (1991).Google Scholar
  109. 109.
    G. Pfeiffer, Young characters on Coxeter basis elements ofIwahori-Heeke algebras and a Murnaghan-Nakayama formula, J. AIgebm 168 (1994), 525–535.MathSciNetMATHGoogle Scholar
  110. 110.
    G. Pfeiffer, Chamkterwerte von Iwahori-Hecke-Algebren von klassischem Typ, Dissertation,Aachener Beitrage zur Mathematik, Band14, Aachen, 1995.Google Scholar
  111. 111.
    G. Pfeiffer, Character values of Iwahori-Heeke algebras of type B, in: Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 195– 249. Birkhauser, Basel, 1997.Google Scholar
  112. 112.
    A. Ram, A Frobenius formula for the characters of the Heeke algebras, Invent. Math. 106 (1991), 461–488.MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    M.J. Richards, Some decomposition numbers for Heeke algebras of general linear groups, Proc. Camb. Phil. Soc. 119 (1996).Google Scholar
  114. 114.
    H. Rui, On endomorphism algebras arising from multiparameter Heeke algebras, J. Algebra 195 (1997), 308–319.MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    K. -D. Schewe, Blocke exzeptioneller Chevalley-Gruppen, Dissertation, Bonner Mathematische Schriften, nr 165, Bonn, 1985.Google Scholar
  116. 116.
    M. Schonert et al., GAP - Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, fourth ed., 1994.Google Scholar
  117. 117.
    T. Shoji, Character sheaves and almost characters ofreductive groups I, II, Adv. in Math. 111 (1995), 244–313 and 314–354.Google Scholar
  118. 118.
    S. VEIGNEAU, ACE - An algebraic combinatorics environment for the computer algebra system MAPLE, User’s reference manual, Version 2.0, Institut Gaspard Monge (1996), http://veyl.univ-mlv.fr/’’’’sv/ACEGoogle Scholar
  119. 119.
    M.-F. Vigneras, Sur la conjecture locale de Langlands pour GL(n,F) sur Pi, C. R. Acad. Sci. Paris Ser. I Math. 318 (1994), 905–908.MathSciNetMATHGoogle Scholar
  120. 120.
    K. Waki, A note on the decomposition numbers of Sp(4, q), J. Algebra 185 (1996), 105–112.MathSciNetCrossRefGoogle Scholar
  121. 121.
    D. White, Decomposition numbers of Sp(4,q) for primes dividing q±l, J. Algebra 132 (1990), 488–500.MathSciNetMATHCrossRefGoogle Scholar
  122. 122.
    E. Wings, Uber die unipotenten Charaktere der Chevalley-Gruppen vom Typ F4 in guter Charakteristik Dissertation, RWTH Aachen, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Frauke M. Bleher
    • 1
  • Wolfgang Kimmerle
    • 2
  • Klaus W. Roggenkamp
    • 2
  • Martin Wursthor
    • 2
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Mathematisches Institut BUniversität StuttgartStuttgartGermany

Personalised recommendations