Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Loewy series of certain indecomposable modules for Frobenius subgroups


Author: Zong Zhu Lin
Journal: Trans. Amer. Math. Soc. 332 (1992), 391-409
MSC: Primary 20G05; Secondary 20G10
DOI: https://doi.org/10.1090/S0002-9947-1992-1052908-4
MathSciNet review: 1052908
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic $ p > 0$. By studying the translation functors from $ p$-singular weights to $ p$-regular weights, we obtain some results on Loewy series of certain indecomposable modules.


References [Enhancements On Off] (What's this?)

  • [1] H. H. Andersen, An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 60 (1986), 125-153. MR 840301 (87j:22025)
  • [2] H. H. Andersen and M. Kaneda, Loewy series of modules for the first Frobenius kernel in a reductive algebraic group, Proc. London Math. Soc. (3) 59 (1989), 74-98. MR 997252 (90h:20062)
  • [3] E. Cline, B. Parshall, and L. Scott, Algebraic stratification in representation categories, J. Algebra 117 (1988), 504-521. MR 957457 (90d:18004)
  • [4] -, Duality in highest weight categories, Classical Groups and Related Topics, Contemp. Math., vol. 82, Amer. Math. Soc., Providence, R. I., 1989, pp. 7-22. MR 982273 (90g:17014)
  • [5] S. Donkin, Rational representations of algebraic groups, Lecture Notes in Math., vol. 1140, Springer-Verlag, Berlin and New York, 1985. MR 804233 (87b:20054)
  • [6] -, Skew modules for reductive groups, J. Algebra 113 (1988), 465-479. MR 929773 (89g:20070)
  • [7] S. R. Doty, Character formulas and Frobenius subgroups of algebraic groups, J. Algebra 125 (1989), 331-347. MR 1018950 (90i:20041)
  • [8] T. J. Enright and B. Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc., vol. 67, no. 367, 1987. MR 888703 (88f:22052)
  • [9] R. S. Irving, Projective modules in the category $ {\mathcal{O}_S}$: Loewy series, Trans. Amer. Math. Soc. 291 (1985), 733-754. MR 800260 (87h:17007)
  • [10] -, The socle filtration of a Verma module, Ann. Sci. École Norm. Sup. 21 (1988), 47-65. MR 944101 (89h:17015)
  • [11] -, Loewy filtrations of Weyl modules, Trans. Amer. Math. Soc. (to appear).
  • [12] -, Singular blocks of the category $ \mathcal{O}$, Math. Z. 204 (1990), 209-224. MR 1055986 (91i:17016)
  • [13] J. C. Jantzen, Representations of algebraic groups, Academic Press, Orlando, 1987. MR 899071 (89c:20001)
  • [14] M. Kaneda, Extensions of modules for infinitesimal algebraic groups, J. Algebra 122 (1989), 188-210. MR 994943 (90c:20049)
  • [15] M. Koppinen, On the translation functors for a semisimple algebraic group, Math. Scand. 51 (1982), 217-226. MR 690527 (84f:20048b)
  • [16] -, Homomorphisms between neighboring Weyl modules, J. Algebra 103 (1984), 302-319. MR 860709 (87m:20116)
  • [17] Z. Lin, The structure of cohomology of line bundles on the flag varieties for some groups of rank $ 2$, Ph. D. Thesis, Univ. of Massachusetts, 1989.
  • [18] -, Extensions between simple modules for Frobenius kernels, J. Pure Appl. Algebra 72 (1991), 275-294.
  • [19] H. Schubert, Categories, Springer-Verlag, New York and Berlin, 1972. MR 0349793 (50:2286)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20G05, 20G10

Retrieve articles in all journals with MSC: 20G05, 20G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1052908-4
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society