The Leech lattice is the unique even unimodular lattice of dimension 24 having no elements of norm 2. Its automorphism group 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 $\gc$ (for a construction of the latter, see miracle octad generator).

Construction of the Leech lattice

Let $\Omega = \{1,2,\ldots,24\}$ and assume we have constructed the Golay code $\gc$ 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 $\gc$ ;
• $\sum_{i \in\Omega} x_i \equiv 4m \pmod{8}$ .

Properties of the Leech lattice

• 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 $\mbox{Aut}(\Lambda)$ is the largest Conway group $Co_0$ with order $8\,315\,553\,613\,086\,720\,000 = 2^{22} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23$ .

4. The group $Co_0$ acts transitively 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 coordinate frames. Any two distinct vectors in a coordinate frame are either opposite or orthogonal, and are congruent modulo $2\Lambda$ .

5. Any vector in $\Lambda$ is congruent 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 of $\Lambda / 2 \Lambda$ : $$1 + |\Lambda(2)|/2 + |\Lambda(3)|/2 + |\Lambda(4)|/48 = 2^{24} = |\Lambda / 2 \Lambda|.$$

Bibliography

1

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.