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

 

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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09700-1
PII: S 0002-9939(08)09700-1
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.