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 , we will consider scalar-removed representations , where is a direct product of special linear algebraic groups and is the representation defined naturally by and a dimension vector . In this paper, we give a necessary and sufficient condition on that has only finitely many -orbits. This condition can be paraphrased as a condition concerning lattices of small rank spanned by positive roots of . 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:
https://doi.org/10.1090/S0002-9939-08-09700-1

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.