The 290 fixed-point sublattices of the Leech lattice

Abstract

We determine the orbits of fixed-point sublattices of the Leech lattice with respect to the action of the Conway group Co 0 . There are 290 such orbits. Detailed information about these lattices, the corresponding coinvariant lattices, and the stabilizing subgroups, is tabulated in several tables.

Some comments on symmetric orbifolds of K3

“…Therefore, the multiplicity of the +1-eigenvalue is also even. We stress that this argument has only implications for cyclic subgroups of Co 0 ; non-cyclic subgroups of Co 0 fixing a sublattice of rank exactly 3 do exist, see for example [24]. symmetries with Frame shape π g up to Stab + (v) transformations are in one to one correspondence with the double cosets 21 ).…”

“…Triples (G, ΛG, ΛG) whereG ⊂ Co 0 is the stabilizer of a sublattice of Λ, and ΛG and ΛG = (ΛG) ∩ Λ are the corresponding invariant and coinvariant lattices, were classified in [24] up to isomorphisms. Out of the 290 groups listed in [24], 221 have an invariant sublattice ΛG of rank at least 3. As expected from the analysis of section 2.3, the quantum symmetry of symmetric orbifolds is not included in the group G Π classified in this theorem.…”

“…As a matter of fact, all such classes actually stabilize a lattice of dimension at least 4, so that d ≥ 1. 7 Furthermore, for any two distinct Co 0 classes, the corresponding elements have different sets of eigenvalues, and non-isomorphic lattices Λ g ′ = (Λ g ′ ) ⊥ ∩ Λ (see for example [24,27]). The sets of eigenvalues are most easily encoded in the Frame shape of g ′ , i.e.…”

“…In [24], Höhn and Mason provide the genus of the sublattice Leech g of the positive definite 11 Leech lattice Leech fixed by some g ∈ Co 0 . The genus is given using the Table 2: For each Frame shape π g , we provide: the number of the smallest group containing g, as listed by Höhn and Mason in [24]; the genus of the fixed sublattice Leech g , using the conventions of [24]; and the opposite −q Leech g of the discriminant form of Leech g in Miranda and Morrison's notation [29,30].…”

“…In the following table, we provide for each Frame shape π g , the genus of the fixed sublattice Leech g in the (positive definite) Leech lattice (as reported in [24]), the number of the corresponding group in the list in [24], and the opposite −q Leech g of the discriminant form of Leech g . This will be useful for computing the data in…”