Let be a commutative ring in which is a unit, and let be an latex rm = mr$. Recall that is the set of all tensors in having two consecutive entries equal, and , where acts on by permuting entries.
Prove that for all . Use this to prove that .
Let . We have , as desired.
Suppose ; say the and components of are equal. Note that , and in particular . In the equation , we can break up the right hand side as a summation over the cosets of , each of which is 0. Thus , and we have .
Now suppose . From the equality proved above, we have . Note that for each . (It suffices to notice this is true for adjacent transpositions- i.e. .) Thus .