TITLE: When can we cancel vector bundles from tensor products?
QUESTION [30 upvotes]: Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is:
Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$? 
Some easy answers (if I am not mistaken): one can when $n=1$ or  when $G$ is a line bundle. At this point I am mostly interested in the case when $E$ is a direct sum of line bundles, but any comments/reference/solutions/analogues about other cases would be appreciated.

REPLY [14 votes]: According to this preprint, over a connected proper algebraic variety $X$ there is a universal reductive group $G$ such that isomorphism classes of vector bundles of rank $n$ are in bijection with isomorphism classes of $n$-dimensional representations of $G$. Furthermore the component group is $\pi_1(X)$.
So the cancellation property you seek holds for any simply-connected proper algebraic variety, because it holds for representations of any connected reductive group. We can see this by characters - a representation is uniquely determined by its character, and because the group is connected, there are no zero-divisors in the ring of characters.
This certainly includes $\mathbb P^n$.