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Equal Characteristic

  • Cédric BonnaféEmail author
Chapter
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Part of the Algebra and Applications book series (AA, volume 13)

Abstract

In this chapter we study the representations of the group G in equal, or natural, characteristic. A significant part of this chapter will be dedicated to the construction of the simple kG-modules. This classical construction generalises to the case of finite reductive groups. It turns out that the simple kG-modules are the restrictions of simple “rational representations” of the algebraic group \(\mathbf{G}=\mathrm{SL}_{2}(\mathbb{F})\). Having obtained this description the determination of the decomposition matrices is straightforward. We then determine the (very simple) partition into blocks as well as the Brauer correspondents. There is only one block of defect 1 (which corresponds to the Steinberg character St G ) and the two other \(\mathcal{O}\)-blocks (which both have U as their defect group — recall that the normaliser of U is B) are only distinguished by the action of the centre Z. As the group U is abelian Broué’s conjecture predicts an equivalence of derived categories between the \(\mathcal{O}\)-blocks of G and their Brauer correspondent. This result was shown by Okuyama (Derived equivalences in SL(2,q), 2000), (Remarks on splendid tilting complexes, 2000), but the proof is too difficult to be included in this book. To finish off, if q=p then U is cyclic and we determine the Brauer trees of the two blocks with defect group U.

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References

  1. [Oku1]
    T. Okuyama, Derived equivalences in SL(2,q), preprint (2000). Google Scholar
  2. [Oku2]
    T. Okuyama, Remarks on splendid tilting complexes, dans Representation theory of finite groups and related topics (Kyoto, 1998), Surikaisekikenkyusho Kokyuroku 1149 (2000), 53–59. Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Institut de Mathématiques et de Modélisation de MontpellierCNRS (UMR 5149), Université Montpellier 2Montpellier CedexFrance

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