Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2507.10414 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908516186324992 |
|---|---|
| author | Zheng, Zhiwei |
| author_facet | Zheng, Zhiwei |
| contents | 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 hyperkähler 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Lemma on Leech-like Lattices Zheng, Zhiwei Algebraic Geometry Group Theory 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 hyperkähler 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. |
| title | A Lemma on Leech-like Lattices |
| topic | Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2507.10414 |