A Lemma on Leech-like Lattices

TL;DR

This paper provides a complete, conceptual proof of a lemma concerning Leech pairs and their embeddings into the Leech lattice, addressing gaps in the original proof and extending its validity.

Contribution

The authors offer a revised, comprehensive proof of a key lemma about Leech pairs, improving upon previous incomplete and computer-assisted proofs.

Findings

Established a complete proof of the lemma for Leech pairs

Clarified conditions for primitive embedding into the Leech lattice

Addressed previous gaps and provided a conceptual understanding

Abstract

A Leech pair is defined as a pair $(G,S)$, where $S$ is a positive definite even lattice without roots, equipped with a faithful action of a finite group $G$, such that the invariant sublattice of $S$ under the action of $G$ is trivial, and the induced action of $G$ on the discriminant group of $S$ is also trivial. This structure appears naturally when investigating hyperk\"ahler manifolds and the symplectic automorphisms acting on them. An important lemma due to Gaberdiel--Hohenegger--Volpato asserts that a Leech pair $(G,S)$ admits a primitive embedding into the Leech lattice if $rank(S)+\ell(A_S)\le 24$. However, the original proof is incomplete, as demonstrated by a counterexample provided by Marquand and Muller. They also presented a computer-assisted proof of the lemma for cases where $rank(S) \le 21$. In this paper, we modify the original approach to provide a complete and conceptual proof of the lemma.