II25,1
In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector. In particular it is closely related to the Leech lattice Λ, and has the Conway group Co1 at the top of its automorphism group.
Write Rm,n for the m+n dimensional vector space Rm+n with the inner product of (a1,...,am+n) and (b1,...,bm+n) given by
The lattice II25,1 is given by all vectors (a1,...,a26) in R25,1 such that either all the ai are integers or they are all integers plus 1/2, and their sum is even.
The lattice II25,1 can be written as Λ⊕H where H is the 2-dimensional even Lorentzian lattice, generated by 2 norm 0 vectors z and w with inner product –1. So we can write vectors of II25,1 as (λ,m, n) = λ+mz+nw with λ in the Leech lattice and m,n integers, where (λ,m, n) has norm λ2 –2mn.
Conway showed that the roots (norm 2 vectors) having inner product –1 with w=(0,0,1) are the simple roots of the reflection group. These are the vectors (λ,1,λ2/2–1) for λ in the Leech lattice. In other words the simple roots can be identified with the points of the Leech lattice, and moreover this is an isometry from the set of simple roots to the Leech lattice.
The reflection group is a hyperbolic reflection group acting on 25-dimensional hyperbolic space. The fundamental domain of the reflection group has 1+23+284 orbits of vertices as follows:
Conway (1983) described the automorphism group Aut(II25,1) of II25,1 as follows.
Every non-zero vector of II25,1 can be written uniquely as a positive integer multiple of a primitive vector, so to classify all vectors it is sufficient to classify the primitive vectors.
Any two positive norm primitive vectors with the same norm are conjugate under the automorphism group.
There are 24 orbits of primitive norm 0 vectors, corresponding to the 24 Niemeier lattices. The correspondence is given as follows: if z is a norm 0 vector, then the lattice z⊥/z is a 24-dimensional even unimodular lattice and is therefore one of the Niemeier lattices.
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