TITLE: Exterior power of a tensor product
QUESTION [14 upvotes]: Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. 
Is there some simple way to decompose this into tensor products of various exterior powers of individual bundles?
I am interested in the case when $F$ corresponds to the twisted line bundles $\mathcal{O}(k)$.

REPLY [3 votes]: There is also a nice description in terms of Schur functor and Young diagrams.
A Young diagram $\lambda$ is a picture made by a finite set of cells, left-aligned in rows, such that the length of the rows decreases going down. Any Young diagram can be transposed (exchanging rows and columns) to obtain another Young diagram $\lambda'$.
To any Young diagram you can associate a functor $S_\lambda$, called Schur functor, which is an endofunctor of the category of finite vector spaces over a fixed field. The way how the Schur functor is constructed starting from the Young diagram is a rather complicated and you can find the details in this page of ncatlab. 
To make an example, the Schur functor associated to a diagram made by only one row with $n$ cells is the functor sending any vector space $V$ to the vector space of symmetric $n$-powers $Sym^n(V)$, while the transposed diagram gives the exterior $n$-powers $\Lambda^n(V)$.
The product of two Schur functors can be decomposed into the linear combination of other Schur functors thanks to the Littlewood–Richardson rule.
The application of this rule in the case of the $n$th exterior power of a tensor product yields the following formula:
$\Lambda^n(V \otimes W) = \bigoplus (S_\lambda(V) \otimes S_{\lambda'}(W))$
where $\lambda$ in the direct sum runs over al the Young diagrams with $n$ cells and at most dim($V$) rows and dim($W$) columns.
You can find a reference for this formula in Fulton, Harris, Representation Theory, exercise 6.11.
[edit: this is an expaned comment based on the hint given in this mathoverflow discussion:
mathoverflow.net/questions/126219/exterior-and-symmetric-powers-of-external-tensor-products-of-representations]