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I interpret the question as follows. Let $SU(3)$ be the group of complex $3\times 3$ matrices with $U^\dagger U=I$ and $\det U=1$ acting as linear operators in $\mathbb C^3$. Is there a subspace $M \subset \mathbb C^3$ with $M \neq \{0\}$, $M \neq \mathbb C^3$ such that $U(M) \subset M$ for every $U \in SU(3)$? In other words, is the natural action of ...
I) The main point is that we usually only consider tensor products $V \otimes W$ of vector spaces $V$, $W$; but groups (say $G$, $H$) are often not vector spaces. If we only consider tensor products of vector spaces, then the object $G \otimes H$ is nonsense, mathematically speaking. With further assumptions on the groups $G$ and $H$, it is sometimes ...