On prehomogeneity of a rank variety

Ouchi, Masaya; Hamada, Michio; Kimura, Tatsuo

doi:10.1090/S0002-9939-2012-11525-4

On prehomogeneity of a rank variety
HTML articles powered by AMS MathViewer

by Masaya Ouchi, Michio Hamada and Tatsuo Kimura PDF

Proc. Amer. Math. Soc. 140 (2012), 4127-4129 Request permission

Abstract:

If a linear algebraic group $G$ acts on $M(m,n)$, then it also acts on a rank variety $M^{(r)}(m,n)=\{ X\in M(m,n)|\ \textrm {rank} X=r\}$. In this paper, we give the necessary and sufficient condition that this variety has a Zariski-dense $G$-orbit. We consider everything over the complex number field $\mathbb {C}.$

References

Tatsuo Kimura, Introduction to prehomogeneous vector spaces, Translations of Mathematical Monographs, vol. 215, American Mathematical Society, Providence, RI, 2003. Translated from the 1998 Japanese original by Makoto Nagura and Tsuyoshi Niitani and revised by the author. MR 1944442, DOI 10.1090/mmono/215
Tatsuo Kimura, Shin-ichi Kasai, Masanobu Taguchi, and Masaaki Inuzuka, Some P.V.-equivalences and a classification of $2$-simple prehomogeneous vector spaces of type $\textrm {II}$, Trans. Amer. Math. Soc. 308 (1988), no. 2, 433–494. MR 951617, DOI 10.1090/S0002-9947-1988-0951617-6
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR 430336