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Support varieties, AR-components, and good filtrations

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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 G r T-modules, we determine the position of simple modules and baby Verma modules within the stable Auslander–Reiten quiver Γ s (G r ) 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.

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References

  1. Alperin J., Evens L.: Representations, resolutions and Quillen’s dimension theorem. J. Pure Appl. Algebra 22, 1–9 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auslander M., Reiten I., Smalø S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  3. Benson D.: Representations and Cohomology, I. Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  4. Benson D.: Representations and Cohomology, II. Cambridge Studies in Advanced Mathematics, vol. 31. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  5. Bessenrodt C.: Modular representation theory for blocks with cyclic defect groups via the Auslander–Reiten quiver. J. Algebra 140, 247–262 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bessenrodt C.: The Auslander–Reiten quiver of a modular group algebra revisited. Math. Z. 206, 25–34 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bruns W., Herzog J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  8. Carlson, J.: Complexity and Krull Dimension. Lecture Notes in Mathematics, vol. 903, pp. 62–67. Springer, Berlin (1981)

  9. Carlson J.: The variety of an indecomposable module is connected. Invent. Math. 77, 291–299 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cline E., Parshall B., Scott L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)

    MATH  MathSciNet  Google Scholar 

  11. Cline E., Parshall B., Scott L., van der Kallen W.: Rational and generic cohomology. Invent. Math. 39, 143–163 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  12. Demazure M., Gabriel P.: Groupes Algébriques. Masson/North-Holland, Paris/Amsterdam (1970)

    MATH  Google Scholar 

  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. Erdmann K., Skowroński A.: On Auslander–Reiten components of blocks and self-injective special biserial algebras. Trans. Am. Math. Soc. 330, 165–189 (1992)

    Article  MATH  Google Scholar 

  15. Farnsteiner R.: Periodicity and representation type of modular Lie algebras. J. Reine Angew. Math. 464, 47–65 (1995)

    MATH  MathSciNet  Google Scholar 

  16. Farnsteiner R.: On support varieties of Auslander–Reiten components. Indag. Math. 10, 221–234 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Farnsteiner R.: Auslander–Reiten components for Lie algebras of reductive groups. Adv. Math. 155, 49–83 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Farnsteiner R.: On the Auslander–Reiten quiver of an infinitesimal group. Nagoya Math. J. 160, 103–121 (2000)

    MATH  MathSciNet  Google Scholar 

  19. Farnsteiner R.: Auslander–Reiten components for G 1 T-modules. J. Algebra Appl. 4, 739–759 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Farnsteiner R.: Group-graded algebras, extensions of infinitesimal groups, and applications. Transform. Groups 14, 127–162 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Farnsteiner, R.: Complexity, periodicity and one-parameter subgroups (in preparation)

  22. Farnsteiner R., Röhrle G.: Almost split sequences of Verma modules. Math. Ann. 322, 701–743 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Fischman D., Montgomery S., Schneider H: Frobenius extensions of subalgebras of Hopf algebras. Trans. Am. Math. Soc. 349, 4857–4895 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Friedlander E., Parshall B.: Support varieties for restricted Lie algebras. Invent. Math. 86, 553–586 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  25. Friedlander E., Suslin A.: Cohomology of finite group schemes over a field. Invent. Math. 127, 209–270 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gordon R., Green E.: Graded Artin algebras. J. Algebra 76, 111–137 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gordon R., Green E.: Representation theory of graded Artin algebras. J. Algebra 76, 138–152 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  28. Happel D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. LMS Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)

    Book  Google Scholar 

  29. Heller A.: Indecomposable representations and the loop space operation. Proc. Am. Math. Soc. 12, 640–643 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  30. Humphreys J.: Symmetry for finite dimensional Hopf algebras. Proc. Am. Math. Soc. 68, 143–146 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. Humphreys J.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, Berlin (1981)

    Google Scholar 

  32. Jantzen J.: Über Darstellungen höherer Frobenius–Kerne halbeinfacher algebraischer Gruppen. Math. Z. 164, 271–292 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  33. Jantzen J.: Support varieties of Weyl modules. Bull. Lond. Math. Soc. 19, 238–244 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  34. Jantzen J.: Modular representations of reductive groups. J. Pure Appl. Algebra 152, 133–185 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jantzen J.: Representations of Algebraic Groups, 2nd edn. Mathematical Surveys and Monographs, vol. 107. American Mathematical Society, Providence (2003)

    Google Scholar 

  36. Jost T.: On Specht modules in the Auslander–Reiten quiver. J. Algebra 173, 281–301 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kawata S.: On Auslander–Reiten components and simple modules for finite group algebras. Osaka J. Math. 34, 681–688 (1997)

    MATH  MathSciNet  Google Scholar 

  38. Kawata S., Michler G., Uno K.: On simple modules in the Auslander–Reiten components of finite groups. Math. Z. 234, 375–398 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  40. Larson R., Sweedler M.: An associative orthogonal form for Hopf algebras. Am. J. Math. 91, 75–94 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  41. Nakano D., Parshall B., Vella D.: Support varieties for algebraic groups. J. Reine Angew. Math. 547, 15–49 (2002)

    MATH  MathSciNet  Google Scholar 

  42. Okuyama T.: On the Auslander–Reiten quiver of a finite group. J. Algebra 110, 425–430 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  43. Pfautsch W.: Ext1 for the Frobenius kernels of SL2. Comm. Algebra 13, 169–179 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  44. Riedtmann C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55, 199–224 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  45. Ringel C.: Finite-dimensional hereditary algebras of wild representation type. Math. Z. 161, 235–255 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  47. Reiten I.: Almost split sequences for group algebras of finite representation type. Trans. Am. Math. Soc. 233, 125–136 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  48. Schue J.: Symmetry for the enveloping algebra of a restricted Lie algebra. Proc. Am. Math. Soc. 16, 1123–1124 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  49. Springer T.: Linear Algebraic Groups. Progress in Mathematics, vol. 9. Birkhäuser, Boston (1998)

    Book  Google Scholar 

  50. Suslin A., Friedlander E., Bendel C.: Infinitesimal 1-parameter subgroups and cohomology. J. Am. Math. Soc. 10, 693–728 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  51. Suslin A., Friedlander E., Bendel C.: Support varieties for infinitesimal group schemes. J. Am. Math. Soc. 10, 729–759 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  52. Uno K.: On the vertices of modules in the Auslander–Reiten quiver. Math. Z. 208, 411–436 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  53. Waterhouse W.: Introduction to Affine Group Schemes. Graduate Texts in Mathematics, vol. 66. Springer, Berlin (1979)

    Google Scholar 

  54. Weibel C.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  55. Webb P.: The Auslander–Reiten quiver of a finite group. Math. Z. 179, 97–121 (1982)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Rolf Farnsteiner

  2. Department of Mathematics, University of Bochum, 44780, Bochum, Germany

    Gerhard Röhrle

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  1. Rolf Farnsteiner
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  2. Gerhard Röhrle
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Correspondence to Rolf Farnsteiner.

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Farnsteiner, R., Röhrle, G. Support varieties, AR-components, and good filtrations. Math. Z. 267, 185–219 (2011). https://doi.org/10.1007/s00209-009-0616-6

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