A Lemma on Leech-like Lattices

2025/07/14 by Zheng, Zhiwei
#Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.10414

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 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)+ℓ(A_S)≤ 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) ≤ 21. In this paper, we modify the original approach to provide a complete and conceptual proof of the lemma.

Citations

• On birational automorphisms of double EPW-cubes
• Finite symplectic automorphism groups of supersingular K3 surfaces
• Borcherds lattices and K3 surfaces of zero entropy
• Finite groups of symplectic automorphisms of hyperkähler manifolds of type K3^[2]
• On derived categories of K3 surfaces, symplectic automorphisms and the\n Conway group

Related

• A note on construction of the Leech lattice
• Rootless pairs of EE₈-lattices
• The Leech lattice and other lattices
• Holes of the Leech lattice and the projective models of K3 surfaces
• Finite groups of symplectic automorphisms of hyperkähler manifolds of type K3^[2]