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Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It is well-known that the Elements $v^{\odot k}$ for $v\in V$ generate this space (see, for instance, this answer on math.se), so they must contain a basis.

In other words, let $N=\binom{n+k-1}k$, then there must be $v_1,\ldots,v_N\in V$ with $$\Sym^k V = \mathbb Cv_1^{\odot k} \oplus \cdots \oplus \mathbb C v_N^{\odot k}.$$ I am looking for an explicit description of such a basis. Is such a description known? Is there maybe even a "nice" or somewhat "natural" choice for the $v_i$?

As pointed out by Martin Brandenburg, I certainly allow choosing a basis $x_1,\ldots,x_n\in V$ and expressing the $v_i$ as linear combinations of these.

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Can you give a reference for the result that the $v^{\odot k}$ generate the symmetric power? Also the $v_i$ should depend on a basis of $V$, right? –  Martin Brandenburg Apr 10 '13 at 9:23
    
@MartinBrandenburg: Good point(s), I edited the question accordingly. –  Jesko Hüttenhain Apr 10 '13 at 9:29

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