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)

 
 

 

First cohomology for finite groups of Lie type: Simple modules with small dominant weights


Author: University of Georgia VIGRE Algebra Group
Journal: Trans. Amer. Math. Soc. 365 (2013), 1025-1050
MSC (2010): Primary 20G10; Secondary 20G05
DOI: https://doi.org/10.1090/S0002-9947-2012-05664-9
Published electronically: October 12, 2012
MathSciNet review: 2995382
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ k$ be an algebraically closed field of characteristic $ p > 0$, and let $ G$ be a simple, simply connected algebraic group defined over $ \mathbb{F}_p$. Given $ r \geq 1$, set $ q=p^r$, and let $ G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $ \operatorname {H}^1(G(\mathbb{F}_q),L(\lambda ))$, where $ L(\lambda )$ is the simple $ G$-module of highest weight $ \lambda $. Under certain very mild conditions on $ p$ and $ q$, we are able to completely describe the first cohomology group when $ \lambda $ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $ \lambda $ is a minimal non-zero dominant weight.


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

  • 1. Henning Haahr Andersen, Extensions of modules for algebraic groups, Amer. J. Math. 106 (1984), no. 2, 489-504. MR 737781 (86g:20056)
  • 2. Michael Aschbacher and Robert M. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), 446-460. MR 0760022 (86m:20060)
  • 3. Christopher P. Bendel, Daniel K. Nakano, and Cornelius Pillen, On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels, J. Pure Appl. Algebra 163 (2001), no. 2, 119-146. MR 1846657 (2002e:20094)
  • 4. Edward Cline, Brian Parshall, and Leonard Scott, Cohomology of finite groups of Lie type. I, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 169-191. MR 0399283 (53:3134)
  • 5. Edward Cline, Brian Parshall, Leonard Scott, and Wilberd van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), no. 2, 143-163. MR 0439856 (55:12737)
  • 6. Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications. MR 1144017 (93i:20059)
  • 7. Eric M. Friedlander, Weil restriction and support varieties, J. Reine Angew. Math. 648 (2010), 183-200. MR 2774309
  • 8. Eric M. Friedlander and Brian J. Parshall, Cohomology of infinitesimal and discrete groups, Math. Ann. 273 (1986), no. 3, 353-374. MR 824427 (87e:22026)
  • 9. The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4.12, 2008.
  • 10. Peter B. Gilkey and Gary M. Seitz, Some representations of exceptional Lie algebras, Geom. Dedicata 25 (1988), no. 1-3, 407-416, Geometries and groups (Noordwijkerhout, 1986). MR 925845 (89h:20056)
  • 11. Robert M. Guralnick, The dimension of the first cohomology group, Representation theory, II (Ottawa, Ont., 1984), Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 94-97. MR 842479 (87h:20095)
  • 12. Robert M. Guralnick and Corneliu Hoffman, The first cohomology group and generation of simple groups, Groups and Geometry (Sienna 1996), Trends. Math., Birkhauser, Basel, 1998, pp. 81-89. MR 1644977 (99h:20087)
  • 13. Robert M. Guralnick and Pham Huu Tiep, First cohomology groups of Chevalley groups in cross characteristic, Ann. of Math. (2) 174 (2011), no. 1, 543-559. MR 2811608.
  • 14. James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978. Second printing, revised. MR 499562 (81b:17007)
  • 15. -, Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, vol. 326, Cambridge University Press, Cambridge, 2006. MR 2199819 (2007f:20023)
  • 16. Jens Carsten Jantzen, First cohomology groups for classical Lie algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 289-315. MR 1112165 (92e:17024)
  • 17. -, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057 (2004h:20061)
  • 18. Wayne Jones, Cohomology of finite groups of Lie type, Ph.D. thesis, University of Minnesota, 1975, ProQuest LLC, Thesis. MR 2625445
  • 19. Wayne Jones and Brian Parshall, On the $ 1$-cohomology of finite groups of Lie type, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) (New York), Academic Press, 1976, pp. 313-328. MR 0404470 (53:8272)
  • 20. A. S. Kleshchev and J. Sheth, On extensions of simple modules over symmetric and algebraic groups, J. Algebra 221 (1999), no. 2, 705-722. MR 1728406 (2001f:20091)
  • 21. -, Corrigendum: On extensions of simple modules over symmetric and algebraic groups, J. Algebra 238 (2001), no. 2, 843-844. MR 1823787
  • 22. Zongzhu Lin and Daniel K. Nakano, Complexity for modules over finite Chevalley groups and classical Lie algebras, Invent. Math. 138 (1999), no. 1, 85-101. MR 1714337 (2000m:20077)
  • 23. Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215 (96d:18001)
  • 24. Daniel K. Nakano, Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and connections, Lie Theory and Representation Theory, Survey of Modern Mathematics Volume II (edited by N. Hu, B. Shu and J.P. Wang), International Press, 2012, pp. 151-176.
  • 25. Brian J. Parshall and Leonard L. Scott, Integral and graded quasi-hereditary algebras, II with applications to representations of generalized $ q$-schur algebras and algebraic groups, 2009.
  • 26. Brian J. Parshall and Leonard L. Scott, Bounding Ext for modules for algebraic groups, finite groups, and quantum groups, Advances in Mathematics 226 (2011), 2065-2088. MR 2739773 (2012b:20109)
  • 27. Leonard L. Scott, Some new examples in 1-cohomology, J. Algebra 260 (2003), no. 1, 416-425, Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976701 (2004f:20092)
  • 28. University of Georgia VIGRE Algebra Group, On Kostant's theorem for Lie algebra cohomology, Representation theory, Contemp. Math., vol. 478, Amer. Math. Soc., Providence, RI, 2009, University of Georgia VIGRE Algebra Group: Irfan Bagci, Brian D. Boe, Leonard Chastkofsky, Benjamin Connell, Bobbe J. Cooper, Mee Seong Im, Tyler Kelly, Jonathan R. Kujawa, Wenjing Li, Daniel K. Nakano, Kenyon J. Platt, Emilie Wiesner, Caroline B. Wright and Benjamin Wyser, pp. 39-60. MR 2513265 (2010i:17029)
  • 29. Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035 (96d:11071)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 20G10, 20G05

Retrieve articles in all journals with MSC (2010): 20G10, 20G05


Additional Information

University of Georgia VIGRE Algebra Group
Affiliation: Department of Mathematics, University of Georgia. Athens, Georgia 30602-7403; Department of Mathematics, Statistics, and Computer Science, University of Wisconsin–Stout, Menomonie, Wisconsin 54751

DOI: https://doi.org/10.1090/S0002-9947-2012-05664-9
Received by editor(s): October 21, 2010
Received by editor(s) in revised form: July 3, 2011, and July 5, 2011
Published electronically: October 12, 2012
Additional Notes: The members of the University of Georgia VIGRE Algebra Group are Brian D. Boe, Adrian M. Brunyate, Jon F. Carlson, Leonard Chastkofsky, Christopher M. Drupieski, Niles Johnson, Benjamin F. Jones, Wenjing Li, Daniel K. Nakano, Nham Vo Ngo, Duc Duy Nguyen, Brandon L. Samples, Andrew J. Talian, Lisa Townsley, and Benjamin J. Wyser.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society