Slopes of Euclidean lattices, tensor product and group actions

TL;DR

This paper investigates how the minimal slope of Euclidean lattices behaves under tensor products, proving the conjecture in specific cases involving automorphism group actions and irreducible components.

Contribution

It proves the conjecture for lattices with automorphism groups acting multiplicity-free and with limited irreducible components, advancing understanding of lattice tensor products.

Findings

Confirmed the conjecture for certain automorphism group actions

Established the minimal slope behavior under specific symmetry conditions

Provided new insights into lattice tensor product properties

Abstract

We study the behaviour of the minimal slope of Euclidean lattices under tensor product. A general conjecture predicts that $\mu_{min}(L \otimes M) = \mu_{min}(L)\mu_{min}(M)$ for all Euclidean lattices $L$ and $M$. We prove that this is the case under the additional assumptions that $L$ and $M$ are acted on multiplicity-free by their automorphism group, such that one of them has at most $2$ irreducible components.