Mathematische Zeitschrift

, Volume 267, Issue 1–2, pp 185–219

Support varieties, AR-components, and good filtrations

Article

Abstract

Let G be a reductive group, defined over the Galois field \({\mathbb{F}_p}\) with p being good for G. Using support varieties and covering techniques based on GrT-modules, we determine the position of simple modules and baby Verma modules within the stable Auslander–Reiten quiver Γs(Gr) of the rth Frobenius kernel of G. In particular, we show that the almost split sequences terminating in these modules usually have an indecomposable middle term. Concerning support varieties, we introduce a reduction technique leading to isomorphisms
$$\mathcal{V}_{G_r}(Z_r(\lambda)) \cong \mathcal{V}_{G_{r-d}}(Z_{r-d}(\mu))$$
for baby Verma modules of certain highest weights \({\lambda, \mu \in X(T)}\), which are related by the notion of depth.

Mathematics Subject Classification (2000)

Primary 16G70 Secondary 17B50 

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References

  1. 1.
    Alperin J., Evens L.: Representations, resolutions and Quillen’s dimension theorem. J. Pure Appl. Algebra 22, 1–9 (1981)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Auslander M., Reiten I., Smalø S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  3. 3.
    Benson D.: Representations and Cohomology, I. Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1991)Google Scholar
  4. 4.
    Benson D.: Representations and Cohomology, II. Cambridge Studies in Advanced Mathematics, vol. 31. Cambridge University Press, Cambridge (1991)Google Scholar
  5. 5.
    Bessenrodt C.: Modular representation theory for blocks with cyclic defect groups via the Auslander–Reiten quiver. J. Algebra 140, 247–262 (1991)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bessenrodt C.: The Auslander–Reiten quiver of a modular group algebra revisited. Math. Z. 206, 25–34 (1991)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bruns W., Herzog J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1998)Google Scholar
  8. 8.
    Carlson, J.: Complexity and Krull Dimension. Lecture Notes in Mathematics, vol. 903, pp. 62–67. Springer, Berlin (1981)Google Scholar
  9. 9.
    Carlson J.: The variety of an indecomposable module is connected. Invent. Math. 77, 291–299 (1984)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cline E., Parshall B., Scott L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)MATHMathSciNetGoogle Scholar
  11. 11.
    Cline E., Parshall B., Scott L., van der Kallen W.: Rational and generic cohomology. Invent. Math. 39, 143–163 (1977)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Demazure M., Gabriel P.: Groupes Algébriques. Masson/North-Holland, Paris/Amsterdam (1970)MATHGoogle Scholar
  13. 13.
    Erdmann K.: Blocks of Tame Representation Type and Related Classes of Algebras. Lecture Notes in Mathematics, vol. 1428. Springer, Berlin (1990)Google Scholar
  14. 14.
    Erdmann K., Skowroński A.: On Auslander–Reiten components of blocks and self-injective special biserial algebras. Trans. Am. Math. Soc. 330, 165–189 (1992)MATHCrossRefGoogle Scholar
  15. 15.
    Farnsteiner R.: Periodicity and representation type of modular Lie algebras. J. Reine Angew. Math. 464, 47–65 (1995)MATHMathSciNetGoogle Scholar
  16. 16.
    Farnsteiner R.: On support varieties of Auslander–Reiten components. Indag. Math. 10, 221–234 (1999)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Farnsteiner R.: Auslander–Reiten components for Lie algebras of reductive groups. Adv. Math. 155, 49–83 (2000)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Farnsteiner R.: On the Auslander–Reiten quiver of an infinitesimal group. Nagoya Math. J. 160, 103–121 (2000)MATHMathSciNetGoogle Scholar
  19. 19.
    Farnsteiner R.: Auslander–Reiten components for G 1 T-modules. J. Algebra Appl. 4, 739–759 (2005)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Farnsteiner R.: Group-graded algebras, extensions of infinitesimal groups, and applications. Transform. Groups 14, 127–162 (2009)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Farnsteiner, R.: Complexity, periodicity and one-parameter subgroups (in preparation)Google Scholar
  22. 22.
    Farnsteiner R., Röhrle G.: Almost split sequences of Verma modules. Math. Ann. 322, 701–743 (2002)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Fischman D., Montgomery S., Schneider H: Frobenius extensions of subalgebras of Hopf algebras. Trans. Am. Math. Soc. 349, 4857–4895 (1997)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Friedlander E., Parshall B.: Support varieties for restricted Lie algebras. Invent. Math. 86, 553–586 (1986)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Friedlander E., Suslin A.: Cohomology of finite group schemes over a field. Invent. Math. 127, 209–270 (1997)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gordon R., Green E.: Graded Artin algebras. J. Algebra 76, 111–137 (1982)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Gordon R., Green E.: Representation theory of graded Artin algebras. J. Algebra 76, 138–152 (1982)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Happel D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. LMS Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar
  29. 29.
    Heller A.: Indecomposable representations and the loop space operation. Proc. Am. Math. Soc. 12, 640–643 (1961)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Humphreys J.: Symmetry for finite dimensional Hopf algebras. Proc. Am. Math. Soc. 68, 143–146 (1978)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Humphreys J.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, Berlin (1981)Google Scholar
  32. 32.
    Jantzen J.: Über Darstellungen höherer Frobenius–Kerne halbeinfacher algebraischer Gruppen. Math. Z. 164, 271–292 (1979)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Jantzen J.: Support varieties of Weyl modules. Bull. Lond. Math. Soc. 19, 238–244 (1987)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Jantzen J.: Modular representations of reductive groups. J. Pure Appl. Algebra 152, 133–185 (2000)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Jantzen J.: Representations of Algebraic Groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence (2003)Google Scholar
  36. 36.
    Jost T.: On Specht modules in the Auslander–Reiten quiver. J. Algebra 173, 281–301 (1995)MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Kawata S.: On Auslander–Reiten components and simple modules for finite group algebras. Osaka J. Math. 34, 681–688 (1997)MATHMathSciNetGoogle Scholar
  38. 38.
    Kawata S., Michler G., Uno K.: On simple modules in the Auslander–Reiten components of finite groups. Math. Z. 234, 375–398 (2000)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Kawata S., Michler G., Uno K.: On Auslander–Reiten components and simple modules for finite groups of Lie type. Osaka J. Math. 38, 21–26 (2001)MATHMathSciNetGoogle Scholar
  40. 40.
    Larson R., Sweedler M.: An associative orthogonal form for Hopf algebras. Am. J. Math. 91, 75–94 (1969)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Nakano D., Parshall B., Vella D.: Support varieties for algebraic groups. J. Reine Angew. Math. 547, 15–49 (2002)MATHMathSciNetGoogle Scholar
  42. 42.
    Okuyama T.: On the Auslander–Reiten quiver of a finite group. J. Algebra 110, 425–430 (1987)MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Pfautsch W.: Ext1 for the Frobenius kernels of SL2. Comm. Algebra 13, 169–179 (1985)MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Riedtmann C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55, 199–224 (1980)MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Ringel C.: Finite-dimensional hereditary algebras of wild representation type. Math. Z. 161, 235–255 (1978)MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Ringel C.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–223 (1991)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Reiten I.: Almost split sequences for group algebras of finite representation type. Trans. Am. Math. Soc. 233, 125–136 (1977)MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Schue J.: Symmetry for the enveloping algebra of a restricted Lie algebra. Proc. Am. Math. Soc. 16, 1123–1124 (1965)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Springer T.: Linear Algebraic Groups. Progress in Mathematics, vol. 9. Birkhäuser, Boston (1998)CrossRefGoogle Scholar
  50. 50.
    Suslin A., Friedlander E., Bendel C.: Infinitesimal 1-parameter subgroups and cohomology. J. Am. Math. Soc. 10, 693–728 (1997)MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Suslin A., Friedlander E., Bendel C.: Support varieties for infinitesimal group schemes. J. Am. Math. Soc. 10, 729–759 (1997)MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Uno K.: On the vertices of modules in the Auslander–Reiten quiver. Math. Z. 208, 411–436 (1991)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Waterhouse W.: Introduction to Affine Group Schemes. Graduate Texts in Mathematics, vol. 66. Springer, Berlin (1979)Google Scholar
  54. 54.
    Weibel C.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)Google Scholar
  55. 55.
    Webb P.: The Auslander–Reiten quiver of a finite group. Math. Z. 179, 97–121 (1982)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KielKielGermany
  2. 2.Department of MathematicsUniversity of BochumBochumGermany

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