On Harder-Narasimhan filtrations and their compatibility with tensor products

TL;DR

This paper develops a metric approach to Harder-Narasimhan filtrations using buildings and establishes a criterion for their compatibility with tensor products, verified in multiple cases including a novel one.

Contribution

It introduces an abstract numerical criterion for compatibility of Harder-Narasimhan filtrations with tensor products, supported by a new verification case.

Findings

Criterion verified in three cases, including a new one.

Buildings attached to modular lattices facilitate a metric approach.

Compatibility of filtrations with tensor products established under the new criterion.

Abstract

We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new.