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Bounding Cohomology for Finite Groups and Frobenius Kernels

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Abstract

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let G σ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible k G(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible k G(σ)-modules \(L,L^{\prime }\), with a similar result holding for G σ . In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.

We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and G σ , and even for the lengths of the underlying PIMs.

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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

    Daniel K. Nakano

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

    Brian J. Parshall & Leonard L. Scott

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

    Cornelius Pillen

  5. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK

    David Stewart

Authors
  1. Christopher P. Bendel
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  2. Daniel K. Nakano
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  3. Brian J. Parshall
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  4. Cornelius Pillen
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  5. Leonard L. Scott
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  6. David Stewart
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Corresponding author

Correspondence to Daniel K. Nakano.

Additional information

Presented by Jon F. Carlson.

Research of the fifth author was supported in part by NSF grant DMS-1001900

Research of the second author was supported in part by NSF grants DMS-1002135 and DMS-1402271

Research of the third author was supported in part by NSF grant DMS-1001900

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Bendel, C.P., Nakano, D.K., Parshall, B.J. et al. Bounding Cohomology for Finite Groups and Frobenius Kernels. Algebr Represent Theor 18, 739–760 (2015). https://doi.org/10.1007/s10468-014-9514-7

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  • DOI: https://doi.org/10.1007/s10468-014-9514-7

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