I am looking for formulas for writing a basis element of $ Sym^k(H) $ as sum of elements of the form $ v^{\otimes k} $ where $ v\in H $. Here $ H $ is a hilbert space and by basis element I mean the image of elements of the form $ e\_{i\_1}\otimes e\_{i\_2}\otimes....\otimes e\_{i\_k} $ under the map $ H^{\otimes k}\to Sym^k(H) $ where $ \{e\_i\}\_{i\in \mathbb{N}} $ is a orthonormal basis of $ H $.


Its easy to prove that they can be written as sum of $ v^{\otimes k} $. And its easy to derive the expressions using roots of unity for small values $ k $. But any reference for where I can find explicit expressions for any $ k $ would be helpful.



Symmetric tensors as sum of powers
There are of course many ways to do that; I like the following formula, for $e\_1,\ldots ,e\_k\in H$ (I write the product in $Sym^k(H)$ as a usual product):


$$e\_1\ldots e\_k=\frac{(-1)^k}{k!} \sum\_{I\subset [1,k]} (-1)^{\# I}(\sum\_{i\in I}e\_i)^k\ .$$
To prove it, take the term of degree $k$ in the equality $$\prod (1-\exp (e\_i))=1-\sum \exp (e\_i)+\sum\_{i<j}\exp (e\_i+e\_j)+\ldots $$


