Let be a field in which is a unit and let be an -vector space. Recall that acts on the tensor power by permuting the components, and that is defined on by . Prove that is the unique largest subspace of on which each permutation acts as multiplication by .
Suppose such that for all . Then , so that .
In particular, any subspace of upon which every permutation acts as scalar multiplication by is contained in .
Note also that acts on as multiplication by since .