I am lacking the knowledge of multi-linear algebra while I come up with the following problems.
Suppose $k$ is a field of characteristic $p>0$. Let $E$ and $F$ be two $k$-vector spaces of dimension $r$ and $s$ respectively.
How to describe $\bigwedge^n(E \otimes F)$? Direct sum? Filtration?
I would like to have explicit answers to the previous question for $n =2, 3$ or even $n = 4$. If $\dim F = 1$, then $\bigwedge^n(E \otimes F) \simeq \bigwedge^n E \otimes \operatorname{T}^n F \simeq \bigwedge^n E \otimes \operatorname{Sym}^n F$. How about $\dim F =2$?
Do we have natural map $\bigwedge^n(E \otimes F) \to \bigwedge^n(E) \otimes \operatorname{Sym}^n(F)$?
I don't like to confuse the symmetric tensors with symmetric algebra, and alternating tensor with exterior powers. So in my definition, $\operatorname{Sym}^n(E)$ and $\bigwedge^n(E)$ are all quotients of $\operatorname{T}^n(E)$, while the space of symmetric tensors and alternating tensors are subspaces of $\operatorname{T}^n(E)$.
It seems that $\operatorname{Sym}(E^\vee)$ is naturally isomorphic to $\Gamma(E)^\vee$, where $\Gamma(E)$ is the divided power algebra. What is correct definition of $\Gamma(E)$? quotient or sub of something? What is the corresponding sub or quotient?
How does taking dual behave under these constructions?
What is the reference for such problems?
Problem 3 solved. Consider the natural map $\operatorname{T}^n(E \otimes F) \xrightarrow{\sim} (\operatorname{T}^n E) \otimes (\operatorname{T}^n F) \to (\bigwedge^n E) \otimes (\operatorname{Sym}^n F)$. Then we can easily see that $\ker\big(\operatorname{T}^n (E \otimes F)\to \bigwedge^n (E \otimes F)\big)$ goes to zero under this natural map, hence it factors through $\bigwedge^n (E \otimes F)$.
3'. Is $(\bigwedge^n E) \otimes (\operatorname{Sym}^n F)$ a sub/quotient/subquotient of $\wedge^n (E\otimes F)$.