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Let $W$ be a real vector space of dimension $2$ and let $\rho_k:GL_2(\mathbb{R}) \to GL(\mathbf{S}^kW)$ be the standard representation of $GL_2(\mathbb{R})$. Since $\rho_k$ is polynomial, it naturally extends to a map $\tilde \rho_k:Mat_2(\mathbb{R}) \to End(\mathbf{S}^kW)$. Denote $Sym_2(\mathbb{R})$ the space of real symmetric $2 \times 2$ matrices. Do we know the dimension of the vector space in $End(\mathbf{S}^kW)$ generated by $\tilde \rho_k(Sym_2(\mathbb{R}))$?

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    @JackSchmidt Please consider converting your comments into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2015-05-05

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