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 $C{o}_0$ (sometimes denoted by $\cdot 0$). The quotient of $C{o}_0$ by its center is called $C{o}_1$, a sporadic simple group.

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

1 Construction of the Leech lattice

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

$$\frac{1}{\sqrt{8}}(x_1,x_2,\mathrm{\dots },x_{24})$$

in ${ℝ}^{24}$ where each $x_i$ is an integer, such that

• For some integer $m$, we have $x_i\equiv x_j\equiv m(mod2)$ for all $i,j\in \mathrm{\Omega }$;

• For any integer $n$, the set of coordinates $\{i\in \mathrm{\Omega }:x_i\equiv n(mod4)\}$ is in ${𝒢}_{24}$;

• ${\sum }_{i\in \mathrm{\Omega }}{x}_i\equiv 4m(mod8)$.

2 Properties of the Leech lattice

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

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

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

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

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

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

3. The automorphism group (http://planetmath.org/EquivalentCode) $\text{Aut}(\mathrm{\Lambda })$ is the largest Conway group $C{o}_0$ with order $\mathrm{8\hspace{0.17em}315\hspace{0.17em}553\hspace{0.17em}613\hspace{0.17em}086\hspace{0.17em}720\hspace{0.17em}000}=2^{22}\cdot 3^9\cdot 5^4\cdot 7^2\cdot 11\cdot 13\cdot 23$.

4. The group $C{o}_0$ acts transitively (http://planetmath.org/LeftAction) on the sets $\mathrm{\Lambda }(2)$, $\mathrm{\Lambda }(3)$, $\mathrm{\Lambda }(4)$. For $n=2,3$, the imprimitivity blocks of the action of $C{o}_0$ on $\mathrm{\Lambda }(n)$ are the sets $\{x,-x\}$ where $x\in \mathrm{\Lambda }(n)$. The imprimitivity blocks of the action of $C{o}_0$ on $\mathrm{\Lambda }(4)$ are sets of 48 vectors called . Any two distinct vectors in a are either or orthogonal, and are congruent (http://planetmath.org/QuotientGroup) modulo $2\mathrm{\Lambda }$.

5. Any vector in $\mathrm{\Lambda }$ is modulo $2\mathrm{\Lambda }$ to a vector in $\mathrm{\Lambda }(n)$ for one of $n=0,2,3,4$. The imprimitivity blocks of the action of $C{o}_0$ on these sets account for all classes (http://planetmath.org/EquivalenceClass) of $\mathrm{\Lambda }/2\mathrm{\Lambda }$:

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

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