Unimodular Lattice - Classification

Classification

For indefinite lattices, the classification is easy to describe. Write Rm,n for the m+n dimensional vector space Rm+n with the inner product of (a₁,...,a_m+n) and (b₁,...,b_m+n) given by

a₁b₁+...+a_mb_m − a_m+1b_m+1 − ... − a_m+nb_m+n.

In Rm,n there is one odd indefinite unimodular lattice up to isomorphism, denoted by

I_m,n,

which is given by all vectors (a₁,...,a_m+n) in Rm,n with all the a_i integers.

There are no indefinite even unimodular lattices unless

m − n is divisible by 8,

in which case there is a unique example up to isomorphism, denoted by

II_m,n.

This is given by all vectors (a₁,...,a_m+n) in Rm,n such that either all the a_i are integers or they are all integers plus 1/2, and their sum is even. The lattice II_8,0 is the same as the E₈ lattice.

Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example I_n,0 in each dimension n less than 8, and two examples (I_8,0 and II_8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32.

In some sense unimodular lattices up to dimension 9 are controlled by E₈, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the Dynkin diagram of the norm 2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice.

Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 (the E₈ lattice), two in dimension 16 (E₈2 and II_16,0), and 24 in dimension 24, called the Niemeier lattices (examples: the Leech lattice, II_24,0, II_16,0+II_8,0, II_8,03). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them.

Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). There is one in dimension 23 (called the short Leech lattice), two in dimension 24 (the Leech lattice and the odd Leech lattice), and Bacher & Venkov (2001) showed that there are 0, 1, 3, 38 in dimensions 25, 26, 27, 28. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots.

The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly.

The following table from (King 2003) gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24.

Beyond 32 dimensions, the numbers increase even more rapidly.