Consider algebraic representations of a reductive group $G$ over a field in characteristic $p$. I even want to allow *potentially disconnected* reductive groups, i.e. $G$ could be a finite group. (However I'm also interested if the behavior in the connected case is different.) 


If $V$ and $W$ are two such representations with $V^{\otimes p}$ ismomorphic to $W^{\otimes p}$, are $V$ and $W$ necessarily isomorphic? 


In characteristic $\neq p$ this is obviously false because you can tensor with a character of order $p$, but my question is in characteristic $p$. 



Are modular representations isomorphic if they're isomorphic after raising to the pth power?
Here's one way of constructing counterexamples for finite groups.


Suppose $M$ is a periodic $kG$-module with period $p$: i.e., the $p$th syzygy $\Omega^pM$ is isomorphic to $M$, but $\Omega M\not\cong M$. Then
$$(\Omega M)^{\otimes p}\cong \Omega^pM\otimes M^{\otimes (p-1)}\cong M^{\otimes p},$$
up to projective direct summands.


If $G$ is a $p$-group, so that all projective $kG$-modules are free, then taking the direct sum of $|G|$ copies of $M$ and of $\Omega M$, and adding a free direct summand to whichever is smaller, we can get two modules $X$ and $Y$ of the same dimension, and then $X^{\otimes p}\cong Y^{\otimes p}$ (since they're isomorphic up to free direct summands and have the same dimension).


For example, take $p=2$ and $G$ the quaternion group $Q\_8$. The trivial module $k$ has period $4$, so $M=k\oplus\Omega^2k$ has period $2$. $M$ has dimension $10$ and $\Omega M$ has dimension $14$, so in this case we can take the direct sum of two copies of each module (rather than $|G|=8$ copies), to get modules $X=k\oplus k\oplus\Omega^2k\oplus\Omega^2k\oplus kG$ and $Y=\Omega k\oplus\Omega k\oplus \Omega^3k\oplus\Omega^3k$ with $X^{\otimes 2}\cong Y^{\otimes 2}$.


Examples of suitable periodic modules for odd $p$ can be found in 


*Carlson, Jon F.*, [**Periodic modules with large periods**](http://dx.doi.org/10.2307/2042990), Proc. Am. Math. Soc. 76, 209-215 (1979). [ZBL0419.20011](https://zbmath.org/?q=an:0419.20011).


