Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A generalization of Pillen's theorem for principal series modules


Author: Yutaka Yoshii
Journal: Proc. Amer. Math. Soc. 140 (2012), 3761-3768
MSC (2010): Primary 20C33
DOI: https://doi.org/10.1090/S0002-9939-2012-11395-4
Published electronically: March 13, 2012
MathSciNet review: 2944716
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a connected, semisimple and simply connected algebraic group defined and split over the finite field of order $ p$. Pillen proved in 1997 that the highest weight vectors of some Weyl $ G$-modules generate the principal series modules as submodules for the corresponding finite Chevalley groups. This result is generalized in this paper.


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

  • 1. H. H. Andersen, A sum formula for tilting modules, J. Pure and Applied Algebra 152 (2000), 17-40. MR 1783982 (2001g:20057)
  • 2. R.W. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. London Math. Soc. 32 (1976), 347-384. MR 0396731 (53:592)
  • 3. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts Math. 9, Springer, 1972. MR 0323842 (48:2197)
  • 4. J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Angew. Math. 317 (1980), 157-199. MR 581341 (82b:20057)
  • 5. J. C. Jantzen, Zur Reduktion modulo $ p$ der Charaktere von Deligne und Lusztig, J. Algebra 70 (1981), 452-474. MR 623819 (82m:20045)
  • 6. J. C. Jantzen, Filtrierungen der Darstellungen in der Hauptserie endlicher Chevalley-Gruppen, Proc. London Math. Soc. (3) 49 (1984), 445-482. MR 759299 (86e:20050)
  • 7. J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Math. Surveys Monogr. 107, Amer. Math. Soc., 2003. MR 2015057 (2004h:20061)
  • 8. O. Mathieu, Filtrations of $ G$-modules, Ann. Sci. Éc. Norm. Super., II. Ser. 23 (1990), 625-644. MR 1072820 (92a:20044)
  • 9. C. Pillen, Reduction modulo $ p$ of some Deligne-Lusztig characters, Arch. Math. 61 (1993), 421-433. MR 1241047 (94i:20026)
  • 10. C. Pillen, Loewy series for principal series representations of finite Chevalley groups, J. Algebra 189 (1997), 101-124. MR 1432367 (98a:20019)
  • 11. F.A. Richen, Modular representations of split $ BN$-pairs, Trans. Amer. Math. Soc. 140 (1969), 435-460. MR 0242972 (39:4299)
  • 12. H. Sawada, A characterization of the modular representations of finite groups with split ($ B$,$ N$)-pairs, Math. Z. 155 (1977), 29-41. MR 0450384 (56:8679)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20C33

Retrieve articles in all journals with MSC (2010): 20C33


Additional Information

Yutaka Yoshii
Affiliation: Department of Liberal Studies, Nara National College of Technology, Yamatokoriyama, Nara, 639-1080, Japan
Email: yyoshii@libe.nara-k.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2012-11395-4
Received by editor(s): April 29, 2011
Published electronically: March 13, 2012
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society