# Representations of $\mathrm{SU}(n)$

I have been given the following representation of $\mathrm{SU}(n)$: Let $V_{k,n} \leq \mathbb{C}[z_1,\dots,z_n]$ be the subspace spanned by the degree-$k$ homogeneous polynomials and define $T_{k,n}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(V_{k,n})$ by setting $T_{k,n}(X)p(z_1,\dots,z_n) = p((z_1,\dots,z_n)X)$. I want to show that if $X$ has all non-zero entries, then the matrix for $T_{k,n}(X)$ has all non-zero entries as well. Does anyone have any suggestions on how to show this?

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Idea: Have you written down the general expression for $p$ and $X$ in components, "sum sum product product", to see what a general $T$-component must look like? If you then sort by a base of polynimials as imputs you might be able to take a look at the columns and moreover, maybe some elements like $\text{det}(X)$ can get rid of and the end result. What have you got? – NikolajK Nov 7 '12 at 12:39