On the decomposition of tensor products of monomial modules for finite 2-groups

Dave Benson conjectured in 2020 that if $G$ is a finite $2$-group and $V$ is an odd-dimensional indecomposable representation of $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$, then the only odd-dimensional indecomposable summand of $V \otimes V^*$ is the trivial representation $\Bbbk$. This would imply that a tensor power of an odd-dimensional indecomposable representation of $G$ over $\Bbbk$ has a unique odd-dimensional summand. Benson has further conjectured that, given such a representation $V$, the function sending a positive integer $n$ to the dimension of the unique odd-dimensional indecomposable summand of $V^{\otimes n}$ is quasi-polynomial. We examine this conjecture for monomial modules, a class of graded representations for the group $\mathbb{Z}/{2^r}\mathbb{Z} \times \mathbb{Z}/{2^s}\mathbb{Z}$ which correspond to skew Young diagrams. We prove the tensor powers conjecture for several modules, giving some of the first nontrivial cases where this conjecture has been verified, and we give conjectural quasi-polynomials for a broad range of monomial modules based on computational evidence.