$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ \sym(\alpha)=\frac{1}{k!}\sum_{\sigma\in S_k}\ ^\sigma\alpha $$ where $S_k$ is the symmetric group on $k$ letters and $^\sigma\alpha$ denotes the action of $\sigma$ on $\alpha$.
What is the kernel of the map $\sym:\bigotimes^k V\to \bigotimes^k V$?
It is clear that all tensors of the form $(u\otimes v-v\otimes u)\otimes\beta$, where $\beta\in \bigotimes^{k-2}V$ are in the kernel. I suspect that these are all the members in the kernel but am unable to prove it.
Can somebody help? Thanks.