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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
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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  • 1. I. Assem, D. Simson, and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts 65, Cambridge University Press, 2006. MR 2197389 (2006j:16020)
  • 2. M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1997. MR 1476671 (98e:16011)
  • 3. P. Gabriel and A. V. Roiter, Representations of finite-dimensional algebras, Springer-Verlag, 1997. MR 1475926 (98e:16014)
  • 4. T. Kamiyoshi, A characterization of finite prehomogeneous vector spaces of $ D_{4}$-type under various scalar restrictions, submitted to Tsukuba J. of Math.
  • 5. T. Kimura, T. Kamiyoshi, N. Maki, M. Ouchi, and M. Takano, A classification of reductive finite prehomogeneous vector spaces of type $ (G\times GL_{n},\, \rho\otimes \varLambda_{1})$ $ (n\geq 2)$ under various restricted scalar multiplications, preprint.
  • 6. T. Kimura, S. Kasai, and O. Yasukura, A classification of the representations of reductive algebraic groups which admit only a finite number of orbits, Amer. J. Math. 108 (1986), no. 3, 643-691. MR 844634 (87k:20074)
  • 7. M. Nagura and T. Niitani, Conditions on a finite number of orbits for $ A\sb r$-type quivers, J. Algebra 274 (2004), no. 2, 429-445. MR 2043357 (2004m:16021)
  • 8. M. Nagura, T. Niitani, and S. Otani, A remark on prehomogeneous actions of linear algebraic groups, Nihonkai Math. J. 14 (2003), no. 2, 113-119. MR 2028468 (2004m:20088)
  • 9. M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1-155. MR 0430336 (55:3341)

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Additional Information

Makoto Nagura
Affiliation: Department of Liberal Studies, Nara National College of Technology, Yamato-Koriyama, Nara 639-1080, Japan

Shin-ichi Otani
Affiliation: School of Engineering, Kanto-Gakuin University, Yokohama, Kanagawa 236-8501, Japan

Daisuke Takeda
Affiliation: Castle Tsuchiura 205, Fujisaki 1–4–6, Tsuchiura, Ibaraki 300-0813, Japan

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.

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