# Leech lattice

The Leech lattice is the unique even (http://planetmath.org/EvenLattice) unimodular lattice of dimension (http://planetmath.org/Dimension2) 24 having no elements of norm 2. Its automorphism group (http://planetmath.org/EquivalentCode) is the largest Conway group $Co_{0}$ (sometimes denoted by $\cdot 0$). The quotient of $Co_{0}$ by its center is called $Co_{1}$, a sporadic simple group.

The construction of the Leech lattice below depends on the existence of the extended binary Golay code $\mathcal{G}_{24}$ (for a construction of the latter, see miracle octad generator).

## 1 Construction of the Leech lattice

Let $\Omega=\{1,2,\ldots,24\}$ and assume we have constructed the Golay $\mathcal{G}_{24}$ on $\Omega$. The Leech lattice $\Lambda$ is the set of all points

 $\frac{1}{\sqrt{8}}(x_{1},x_{2},\ldots,x_{24})$

in $\mathbb{R}^{24}$ where each $x_{i}$ is an integer, such that

• For some integer $m$, we have $x_{i}\equiv x_{j}\equiv m\pmod{2}$ for all $i,j\in\Omega$;

• For any integer $n$, the set of coordinates $\{i\in\Omega:x_{i}\equiv n\pmod{4}\}$ is in $\mathcal{G}_{24}$;

• $\sum_{i\in\Omega}x_{i}\equiv 4m\pmod{8}$.

## 2 Properties of the Leech lattice

1. The Leech lattice $\Lambda$ is an unimodular lattice; in other words:

• The set $\Lambda$ spans all of $\mathbb{R}^{24}$ as an $\mathbb{R}$-vector space.

• For any $x,y\in\Lambda$, the scalar product $x\cdot y$ is an integer.

• For any $x\in\Lambda$, the norm $x\cdot x$ is an even integer.

• The volume of the fundamental parallelogram of $\Lambda$ is 1.

2. Let $\Lambda(n)=\{x\in\Lambda:x\cdot x=2n\}$. Then $|\Lambda(0)|=1$, $|\Lambda(1)|=0$, $|\Lambda(2)|=196560$, $|\Lambda(3)|=16773120$, $|\Lambda(4)|=398034000$.

3. The automorphism group (http://planetmath.org/EquivalentCode) $\mbox{Aut}(\Lambda)$ is the largest Conway group $Co_{0}$ with order $8\,315\,553\,613\,086\,720\,000=2^{22}\cdot 3^{9}\cdot 5^{4}\cdot 7^{2}\cdot 1% 1\cdot 13\cdot 23$.

4. The group $Co_{0}$ acts transitively (http://planetmath.org/LeftAction) on the sets $\Lambda(2)$, $\Lambda(3)$, $\Lambda(4)$. For $n=2,3$, the imprimitivity blocks of the action of $Co_{0}$ on $\Lambda(n)$ are the sets $\{x,-x\}$ where $x\in\Lambda(n)$. The imprimitivity blocks of the action of $Co_{0}$ on $\Lambda(4)$ are sets of 48 vectors called . Any two distinct vectors in a are either or orthogonal, and are congruent (http://planetmath.org/QuotientGroup) modulo $2\Lambda$.

5. Any vector in $\Lambda$ is modulo $2\Lambda$ to a vector in $\Lambda(n)$ for one of $n=0,2,3,4$. The imprimitivity blocks of the action of $Co_{0}$ on these sets account for all classes (http://planetmath.org/EquivalenceClass) of $\Lambda/2\Lambda$:

 $1+|\Lambda(2)|/2+|\Lambda(3)|/2+|\Lambda(4)|/48=2^{24}=|\Lambda/2\Lambda|.$

## References

• 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
 Title Leech lattice Canonical name LeechLattice Date of creation 2013-03-22 18:43:23 Last modified on 2013-03-22 18:43:23 Owner monster (22721) Last modified by monster (22721) Numerical id 7 Author monster (22721) Entry type Definition Classification msc 20D08 Classification msc 20B25 Classification msc 11H56 Classification msc 11H06 Classification msc 51E10 Related topic BinaryGolayCode Related topic MiracleOctadGenerator