Non-negligible summands in tensor powers of some modular representations of finite $p$-groups

Abstract

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all summands of the tensor product $V \otimes V^*$ except for $\Bbbk$ have even dimension. It is known that the analogous result for general $p$ is false. In this paper, we investigate the class of graded representations $V$ which have dimension coprime to $p$ and for which $V \otimes V^*$ has a non-trivial summand of dimension coprime to $p$, for a graded group scheme closely related to $\mathbb{Z}/p^r \mathbb{Z} \times \mathbb{Z}/p^s \mathbb{Z}$, where $r$ and $s$ are nonnegative integers and $p>2$. We produce an infinite family of such representations in characteristic 3 and show in particular that the tensor subcategory generated by any of these representations in the semisimplification contains the modulo $3$ reduction of the category of representations of the symmetric group $S_3$. Our results are compatible with a general version of Benson's conjecture due to Etingof.