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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear maps preserving invariants


Author: Gerald W. Schwarz
Journal: Proc. Amer. Math. Soc. 136 (2008), 4197-4200
MSC (2000): Primary 20G20, 22E46, 22E60
Published electronically: July 23, 2008
MathSciNet review: 2431032
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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$ 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
Email: schwarz@brandeis.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09628-7
PII: S 0002-9939(08)09628-7
Keywords: Invariant polynomials
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.