Leech lattice - Knowlpedia

The Leech lattice $\Lambda\subset\mathbb R^{24}$ is, up to an orthogonal transformation, the unique full-rank Euclidean lattice satisfying

\operatorname{covol}(\Lambda)=1,\qquad \langle x,y\rangle\in\mathbb Z,\qquad \langle x,x\rangle\in2\mathbb Z \quad(x,y\in\Lambda),

and containing no vector $x$ with $\langle x,x\rangle=2$ . Here $\langle-,-\rangle$ is the standard positive-definite inner product, and covolume is the volume of a lattice basis parallelepiped. The integrality, evenness, and covolume-one conditions say that the lattice is even unimodular.

Normalization and symmetry

Its shortest nonzero vectors have squared length $4$ . Thus a “norm-4 vector” in this setting has Euclidean length $2$ ; norm denotes squared length.

Write $O(\Lambda)$ for the group of real linear isometries preserving $\Lambda$ . Its quotient by $\{I,-I\}$ is $\mathrm{Co}_1$ . Stabilizers of vectors with squared lengths $4$ and $6$ give $\mathrm{Co}_2$ and $\mathrm{Co}_3$ , respectively.

References

Henry Cohn and Abhinav Kumar, Optimality and uniqueness of the Leech lattice among lattices, Appendix B, first paragraph (p. 36 in the arXiv PDF), for the uniqueness characterization; §2 for minimal length $2$ at covolume one.
Robert A. Wilson, Finite Simple Groups, Chapter 5 notes, §§5.3.1–5.3.4, especially equations (5.16)–(5.17).