symmetric tensors I am reading a paper ,,The Gelfand map and symmetric product" by V.M. Buchstaber and E.G. Rees 
http://arxiv.org/abs/math/0109122
On the page 6 in the proof of Theorem 2.8 there is considered a subspace $S^n A\subset A^{\otimes n}$ of symmetric tensors. It is written that a typical element of $a\in S^nA$ is of the form $$\textbf{a}=\sum\limits_{\sigma\in S_n}a_{\sigma(1)}\otimes\ldots\otimes a_{\sigma(n)}. $$
I can't understand what $a_{\sigma(i)}$ is and why this equality holds. Thank you very much in advance for any help in understanding this.
 A: I never read that paper, but I am almost sure about the meaning of $$S^nA\subset A^{\otimes n}.$$
First of all: $A^{\otimes n}=A\otimes ... \otimes A$ $n$ times; so an arbitrarily element of $A^{\otimes n}$ can be write in the following way:
$$ \textbf{a}=\sum_{finite} a_1 \otimes ... \otimes a_n,$$ where $a_i \in A$ for every $i=1,...,n$. 
Now $S^nA$ in the subset of symmetric elements in $A^{\otimes n}$; what does symmetric element means?
Consider the action of the group $S_n$ on $A^{\otimes n}$, which act in the following way: let $\sigma \in S_n$ and $\textbf{a}\in A^{\otimes n}$; then 
$$\sigma . \textbf{a}=\sigma.(a_1\otimes...\otimes a_n)=a_{\sigma(1)}\otimes...\otimes a_{\sigma(n)}.$$
In other words every elements in $S_n$ act on an element $\textbf{a}$ permuting the elements $a_i$.

DEFINITION: An element $\textbf{a}$ is called symmetric if: for every
  $\sigma \in S_n$ the following equality holds:
  $$\sigma.\textbf{a}=\textbf{a};$$ and we say that $\textbf{a}\in S^nA$.

Finally is not difficult to see that $\textbf{a}=\sum_{\sigma \in S_n} a_{\sigma(1)}\otimes...\otimes a_{\sigma(n)}\in S^nA$.
Try to see what happens when an element of $S_n$ acts on $\sum_{\sigma \in S_n} a_{\sigma(1)}\otimes...\otimes a_{\sigma(n)}$. 
