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Let $V$ be a vector space of dimension $n$. Let $S^k V$ be a representation of $GL(n)$. I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such that exists $GL(n)$ mapping $T:S^k V \rightarrow V_1\otimes V_2$ such that for some $x$ matrix $T(x)$ is invertible .

Added: Just to clarify: The mapping $T$ itself may not be invertible I am asking that $T(x)$ will be invertible in $V_1\otimes V_2$. For example the mapping $S^2 V \rightarrow V\otimes V$ is not invariable, but $T(x)$ is invartable for most $x$'s.

I am interested for a base field $\mathbb{C}$. And of course this may happen only in case when $S^kV$ is a sub-representation of $V_1 \otimes V_2$. In fact I do not know for which irreps $V_1, V_2$ representation $V_1 \otimes V_2$ have $S^kV$ as a component. I will be happy if you can give me a reference for this question.

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# Representations of $GL(n)$ containing $S^kV$

Let $V$ be a vector space of dimension $n$. Let $S^k V$ be a representation of $GL(n)$. I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such that exists $GL(n)$ mapping $T:S^k V \rightarrow V_1\otimes V_2$ such that for some $x$ matrix $T(x)$ is invertible .