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 $\tilde \rho_k(Sym_2(\mathbb{R}))$?
image of symmetric matrices under representation of $GL_2(\mathbb{R})$
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linear-algebra
matrices
representation-theory
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0Perhaps I'm dense but $\rho_k$ is faithful and therefore so is $\tilde\rho_k$. So the dimension must be three, right? – 2011-06-20
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0@Marek: There is no reason for the map $\rho_k$ to respect addition of matrices --- only products. – 2011-06-20
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0I think it is (k+1)*(k+2)/2. In a fairly standard basis, you get that the strictly upper triangular part is component-wise a multiple of the strictly lower triangular part (different coefficients per component). At least this works for k ≤ 5. – 2011-06-20
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0@Jyrki: I missed the word *generated*. So the question is indeed non-trivial. Thanks. – 2011-06-20
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0@Jack: I agree that it should be (k+1)*(k+2)/2. But I do not see why saying that the upper triangular part is component-wise a multiple of the strictly lower triangular part implies that I get (k+1)*(k+2)/2 linearly independent matrices. – 2011-06-20
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0For the linearly independent side, I just constructed the matrices. I checked it (spanning and independent) up to k ≤ 20. I didn't check if the independent set was easy to specify, but I think the linear dependencies are easy to figure out. – 2011-06-20
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0@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