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 Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be an algebraically closed field of characteristic
, and let
be a simple, simply connected algebraic group defined over
. Given
, set
, and let
be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group
, where
is the simple
-module of highest weight
. Under certain very mild conditions on
and
, we are able to completely describe the first cohomology group when
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
is a minimal non-zero dominant weight.
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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.