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Bounding the Dimensions of Rational Cohomology Groups

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Developments and Retrospectives in Lie Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 38))

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.

Research of Parshall was partially supported by NSF grant DMS 1001900.

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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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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Wisconsin–Stout, Menomonie, WI, 54751, USA

    Christopher P. Bendel

  2. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Brian D. Boe & Daniel K. Nakano

  3. Department of Mathematical Sciences, DePaul University, Chicago, IL, 60614, USA

    Christopher M. Drupieski

  4. Department of Mathematics, University of Virginia, Charlottesville, VA, 22903, USA

    Brian J. Parshall

  5. Department of Mathematics and Statistics, University of South Alabama, Mobile, AL, 36688, USA

    Cornelius Pillen

  6. Louise S. McGehee School, 2343 Prytania Street, New Orleans, LA, 70130, USA

    Caroline B. Wright

Authors
  1. Christopher P. Bendel
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  2. Brian D. Boe
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  3. Christopher M. Drupieski
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  4. Daniel K. Nakano
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  5. Brian J. Parshall
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  6. Cornelius Pillen
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  7. Caroline B. Wright
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Corresponding author

Correspondence to Daniel K. Nakano .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, Santa Cruz, California, USA

    Geoffrey Mason

  2. Jacobs University, Bremen, Germany

    Ivan Penkov

  3. Department of Mathematics, University of California, Berkeley, Berkeley, California, USA

    Joseph A. Wolf

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Bendel, C.P. et al. (2014). Bounding the Dimensions of Rational Cohomology Groups. In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-09804-3_2

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