I was reading Fulton and Harris' discussion of exterior and symmetric products as quotient spaces of tensor products in their rep theory book when I noticed that they made this claim (the emphasis is mine):
The exterior powers $ \bigwedge^n V $ and symmetric powers $ \operatorname{Sym}^n V $ can also be realized as subspaces of $ V^{\otimes n} $, assuming, as we have throughout, that the ground field has characteristic 0.
Why must the ground field be characteristic zero for this to be true? In particular, I don't see any problem with the quotient construction of the exterior and symmetric products for a vector space with nonzero characteristic; if there is no problem with this construction, then the inclusion of these products into the tensor product ought to be well-defined as well. Am I missing something obvious here?
The usual construction of embedding the symmetric power into the tensor power involves dividing by $n!$. This can't always be done in positive characteristic (in fact there are only finitely many $n$ where it is valid).