Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

Question

I have the following question:

Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that the group algebra $k\Sigma_r$ is a selfinjective algebra.

Is it true that $E$ is a faithful $k\Sigma_r$-module for $n \geq r$ (and why?), but not in general for $n<r$ (and why) ?

Thanks for the help!