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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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Perhaps I'm dense but $\rho_k$ is faithful and therefore so is $\tilde\rho_k$. So the dimension must be three, right? – Marek Jun 20 '11 at 9:26
    
@Marek: There is no reason for the map $\rho_k$ to respect addition of matrices --- only products. – Jyrki Lahtonen Jun 20 '11 at 10:33
    
I 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. – Jack Schmidt Jun 20 '11 at 13:56
    
@Jyrki: I missed the word generated. So the question is indeed non-trivial. Thanks. – Marek Jun 20 '11 at 14:58
    
@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. – Max Jun 20 '11 at 17:04

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