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The classification of orbits by a natural action of certain reductive linear groups. (English) Zbl 1161.20306
Summary: When $$n\geq 2m$$, the orbit decomposition of $$\mathbb{C}^n\otimes\mathbb{C}^m$$ by the natural action of $$\text{SO}_n\mathbb{C}\times\text{GL}_m\mathbb{C}$$ was studied more than twenty years ago. In this article, the orbit decomposition is given for any $$n,m\geq 1$$. Then it turns out that there is an orbit which not appeared in the literature when $$n=2m$$.
##### MSC:
 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 20G05 Representation theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients) 17B20 Simple, semisimple, reductive (super)algebras