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

TL;DR

This paper explores the structure of tensor powers of certain modular representations of finite p-groups, providing new examples and insights related to Benson's conjecture and the representation theory of group schemes.

Contribution

It identifies an infinite family of representations in characteristic 3 with non-trivial summands of dimension coprime to p, expanding understanding of tensor decompositions in modular representation theory.

Findings

Constructed an infinite family of such representations in characteristic 3.

Showed the generated tensor subcategory contains the modulo 3 reduction of S_3 representations.

Results align with a generalized Benson's conjecture by Etingof.

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.