# Bounding the Dimensions of Rational Cohomology Groups

## Abstract

Let *k* be an algebraically closed field of characteristic *p* > 0, and let *G* be a simple simply-connected algebraic group over *k*. In this paper we investigate situations where the dimension of a rational cohomology group for *G* can be bounded by a constant times the dimension of the coefficient module. As an application, effective bounds on the first cohomology of the symmetric group are obtained. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.

## Key words

Cohomology Representation theory Algebraic Groups## Mathematics Subject Classification (2010):

20G10.## Notes

### Acknowledgements

The authors thank the American Institute of Mathematics for hosting the workshops “Cohomology and representation theory for finite groups of Lie type” in June 2007, and “Cohomology bounds and growth rates” in June 2012. Many of the results described in Sect. 1.1 were motivated by the ideas exchanged at the 2007 workshop, and provided impetus for the organization of the 2012 meeting. The results of this paper were obtained in the AIM working group format at the 2012 workshop, which promoted a productive exchange of ideas between the authors at the meeting.The fourth author (Nakano) presented talks at the workshop “Lie Groups, Lie Algebras and their Representations” at U.C. Santa Cruz in 1999 and at Louisiana State University (LSU) in 2011. At the LSU meeting, his lecture was devoted to explaining the various connections between the cohomology theories for reductive algebraic groups and their associated Frobenius kernels and finite Chevalley groups. The results in this paper are a natural extension of the results discussed in his presentation.

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