To me, it appears always as a little 'magic' when people reorder expressions, indexed by highly complex combinations of permutations and I would like to know in deep and formally what really is going on.
For example consider the following easy identity:
$ \sum_{\sigma\in S_k}x_{\sigma(1)}\otimes \cdots \otimes x_{\sigma(k)}= \sum_{\sigma\in S_k}x_{\sigma(2)}\otimes x_{\sigma(1)}\otimes \cdots \otimes x_{\sigma(k)} $
where $S_k$ is the set of all permutations of an $k$-element set. Of course this identity is true, since every possible combination is already an element of $S_k$, but how can I really proof the previous equation? Obviously expressions like the previous example can be arbitrary complicated and therefore a formal step by step algorithm would be desireable.
Moreover expressions like the previous one often contain signs and gradings and stuff like that. So a formal method is necessary.
My assumptions are the following:
$\bullet $ The permutation group has a representations on some mathematical structure.
Can someone explain how we formally correct "reorder" i.e. change the indices of expressions indexed by permutations?