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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Linear maps preserving invariants
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by Gerald W. Schwarz PDF
Proc. Amer. Math. Soc. 136 (2008), 4197-4200 Request permission

Abstract:

Let $G\subset \mathrm {GL}(V)$ be a complex reductive group. Let $G’$ denote $\{\varphi \in \mathrm {GL}(V)\mid p\circ \varphi =p\text { for all }p\in \mathbb {C}[V]^G\}$. We show that, “in general”, $G’=G$. In case $G$ is the adjoint group of a simple Lie algebra $\mathfrak {g}$, we show that $G’$ is an order 2 extension of $G$. We also calculate $G’$ for all representations of $\mathrm {SL}_2$.
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Additional Information
  • Gerald W. Schwarz
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454-9110
  • MR Author ID: 157450
  • Email: schwarz@brandeis.edu
  • Received by editor(s): November 14, 2007
  • Published electronically: July 23, 2008
  • Additional Notes: This work was partially supported by NSA Grant H98230-06-1-0023
  • Communicated by: Gail R. Letzter
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 4197-4200
  • MSC (2000): Primary 20G20, 22E46, 22E60
  • DOI: https://doi.org/10.1090/S0002-9939-08-09628-7
  • MathSciNet review: 2431032