# Linear maps preserving orbits

• [1] Brandeis University Department of Mathematics Waltham, MA 02454-9110 (USA)
• Volume: 62, Issue: 2, page 667-706
• ISSN: 0373-0956

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## Abstract

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Let $H\subset GL\left(V\right)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\left\{g\in \phantom{\rule{3.33333pt}{0ex}}\mathrm{GL}\left(V\right)\mid gHv=Hv\right\}$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$-orbits which are not (semi)-characteristic in many cases.

## How to cite

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Schwarz, Gerald W.. "Linear maps preserving orbits." Annales de l’institut Fourier 62.2 (2012): 667-706. <http://eudml.org/doc/251042>.

@article{Schwarz2012,
abstract = {Let $H\subset \operatorname\{GL\}(V)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\lbrace g\in ~\{\rm GL\}(V)\mid gHv = Hv\rbrace$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$-orbits which are not (semi)-characteristic in many cases.},
affiliation = {Brandeis University Department of Mathematics Waltham, MA 02454-9110 (USA)},
author = {Schwarz, Gerald W.},
journal = {Annales de l’institut Fourier},
keywords = {Characteristic orbits; linear preserver problems},
language = {eng},
number = {2},
pages = {667-706},
publisher = {Association des Annales de l’institut Fourier},
title = {Linear maps preserving orbits},
url = {http://eudml.org/doc/251042},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Schwarz, Gerald W.
TI - Linear maps preserving orbits
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 667
EP - 706
AB - Let $H\subset \operatorname{GL}(V)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\lbrace g\in ~{\rm GL}(V)\mid gHv = Hv\rbrace$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$-orbits which are not (semi)-characteristic in many cases.
LA - eng
KW - Characteristic orbits; linear preserver problems
UR - http://eudml.org/doc/251042
ER -

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