Are modular representations isomorphic if they're isomorphic after raising to the pth power?

Question

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 $p$.

Answer 1

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, Proc. Am. Math. Soc. 76, 209-215 (1979). ZBL0419.20011.