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A characterization of finite prehomogeneous vector spaces associated with products of special linear groups and Dynkin quivers


Authors: Makoto Nagura, Shin-ichi Otani and Daisuke Takeda
Journal: Proc. Amer. Math. Soc. 137 (2009), 1255-1264
MSC (2000): Primary 14L30; Secondary 16G20, 11S90
DOI: https://doi.org/10.1090/S0002-9939-08-09700-1
Published electronically: October 22, 2008
MathSciNet review: 2465647
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Abstract: For a given finite-type quiver $\varGamma$, we will consider scalar-removed representations $(S_{d}, R_{d}(\varGamma ))$, where $S_{d}$ is a direct product of special linear algebraic groups and $R_{d}(\varGamma )$ is the representation defined naturally by $\varGamma$ and a dimension vector $d$. In this paper, we give a necessary and sufficient condition on $d$ that $R_{d}(\varGamma )$ has only finitely many $S_{d}$-orbits. This condition can be paraphrased as a condition concerning lattices of small rank spanned by positive roots of $\varGamma$. To determine such scalar-removed representations having only finitely many orbits is very fundamental to the open problem of classification of the so-called semisimple finite prehomogeneous vector spaces. We consider everything over an algebraically closed field of characteristic zero.


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Additional Information

Makoto Nagura
Affiliation: Department of Liberal Studies, Nara National College of Technology, Yamato-Koriyama, Nara 639-1080, Japan
Email: nagura@libe.nara-k.ac.jp

Shin-ichi Otani
Affiliation: School of Engineering, Kanto-Gakuin University, Yokohama, Kanagawa 236-8501, Japan
Email: hocke@kanto-gakuin.ac.jp

Daisuke Takeda
Affiliation: Castle Tsuchiura 205, Fujisaki 1–4–6, Tsuchiura, Ibaraki 300-0813, Japan
Email: d-takeda@f3.dion.ne.jp

Keywords: Semisimple finite prehomogeneous vector space, Dynkin quiver
Received by editor(s): April 22, 2008
Published electronically: October 22, 2008
Communicated by: Martin Lorenz
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.