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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Complexity, periodicity and one-parameter subgroups
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by Rolf Farnsteiner PDF
Trans. Amer. Math. Soc. 365 (2013), 1487-1531 Request permission


Using the variety of infinitesimal one-parameter subgroups introduced by Suslin-Friedlander-Bendel, we define a numerical invariant for representations of an infinitesimal group scheme $\mathcal {G}$. For an indecomposable $\mathcal {G}$-module $M$ of complexity $1$, this number, which may also be interpreted as the height of a “vertex” $\mathcal {U}_M \subseteq \mathcal {G}$, is related to the period of $M$. In the context of the Frobenius category of $G_rT$-modules associated to a smooth reductive group $G$ and a maximal torus $T \subseteq G$, our methods give control over the behavior of the Heller operator of such modules, as well as precise values for the periodicity of their restrictions to $G_r$. Applications include the structure of stable Auslander-Reiten components of $G_rT$-modules as well as the distribution of baby Verma modules.
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Additional Information
  • Rolf Farnsteiner
  • Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
  • MR Author ID: 194225
  • Email:
  • Received by editor(s): February 6, 2011
  • Received by editor(s) in revised form: July 15, 2011
  • Published electronically: October 4, 2012
  • Additional Notes: This work was supported by the D.F.G. priority program SPP1388 ‘Darstellungstheorie’.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1487-1531
  • MSC (2010): Primary 14L15, 16G70; Secondary 16T05
  • DOI:
  • MathSciNet review: 3003272