A conjecture on the tensor ideal for an elementary p-group generated by the restriction of a Steinberg module

TL;DR

This paper investigates a conjecture about tensor products involving Steinberg modules over elementary p-groups, which could impact the classification of tensor categories in positive characteristic.

Contribution

The authors provide evidence supporting a conjecture that tensoring a finite-dimensional representation with a Steinberg module yields a restricted tilting module, with implications for tensor category classification.

Findings

Evidence supporting the conjecture that tensor products produce restricted tilting modules.

Implications for classifying incompressible symmetric tensor categories in positive characteristic.

Potential impact on the understanding of tensor ideals in modular representation theory.

Abstract

In previous work (Coulembier--Flake 2024), the authors conjectured that the tensor product of an arbitrary finite-dimensional modular representation of an elementary abelian $p$-group with the biggest non-projective restricted Steinberg $SL_2$-module is a restricted tilting module. We showed that the validity of the conjecture would have interesting implications in the theory of tensor categories in positive characteristic, in particular, with respect to the classification of incompressible symmetric tensor categories, which is the subject of arguably the main open conjecture in the area. We present here some evidence for the conjecture to hold.