\magnification =\magstep1
Chapter 17
The 24-dimensional odd unimodular lattices.
R. E. Borcherds.
This chapter completes the classification of the 24-dimensional unimodular
lattices by enumerating the odd lattices. These are (essentially) in
one-to-one correspondence with neighboring pairs of Niemeier lattices.
The even unimodular lattices in 24 dimensions were classified by
Niemeier [Nie2] and the results are given in the previous chapter,
together with the enumeration of the even and odd unimodular lattices
in dimensions less than 24. There are twenty-four Niemeier lattices,
and in the present chapter they will be referred to by their components
$D_{24}$, $D_{16}E_8$, with the Leech lattice being denoted by
$\Lambda_{24}$, and also by the Greek letters $\alpha,\beta,\ldots$
(see Table 16.1).
The {\bf odd} unimodular lattices in 24 and 25 dimensions were
classified in [Bor1]. In this chapter we list the odd 24-dimensional
lattices. Only those with minimal norm at least 2 are given, i.e.,
those that are strictly 24-dimensional, since the others can easily be
obtained from lower dimensional lattices (see the summary in Table 2.2
of chapter 2).
There is a table of all the 665 25-dimensional unimodular lattices
and the 121 even 25-dimensional lattices of determinant 2 on my home
page
(currently http://www.dpmms.cam.ac.uk/home/emu/reb/.my-home-page.html).
Two lattices are called {\bf neighbors} if
their intersection has index 2 in each of them ([Kne4], [Ven2]).
We now give a brief description of the algorithm used in [Bor1]
to enumerate the 25-dimensional unimodular lattices.
The first step is to observe that there is a one-to-one correspondence
between 25 dimensional unimodular lattices (up to isomorphism)
and orbits of norm $-4$ vectors in the even Lorentzian lattice $II_{25,1}$
given as follows: the lattice $\Lambda$ corresponds to the norm $-4$
vector $v$ if and only if the sublattice of even vectors of $\Lambda$
is isomorphic to the lattice $v^\perp$.
So we can classify 25 dimensional unimodular lattices if we can
classify negative norm vectors in $II_{25,1}$.
We can classify orbits of vectors of norm $-2n\le 0$ in
$II_{25,1}$ by induction on $n$ as follows. First of all the
primitive norm 0 vectors correspond to the Niemeier lattices as in section 1 of
chapter 26. So there are exactly 24 orbits of primitive norm 0 vectors,
and any norm 0 vector can be obtained from a primitive one by multiplying
by some constant.
Suppose we have classified all orbits of vectors of norms $-2m$ with
$0\ge -2m>-2n$, and that we have a vector $v$ of norm $-2n$. We fix
a fundamental Weyl chamber for the reflection group of
$II_{25,1}$ as in chapter 26. We look at the root system of the lattice
$v^\perp$, and find that one of the following 3 things can happen:
\item{1.} There is a norm 0 vector $z$ with $(z,v)=-1$. It turns out to be
trivial to
classify such norm $-2n$ vectors $v$: there is one orbit corresponding
to each orbit of norm 0 vectors. They correspond to lattices $v^\perp$
which are the sum of a Niemeier lattice and a one dimensional lattice
generated by a vector of norm $2n$.
\item{2.} There is no norm 0 vector $z$ with $(z,v)=1$ and the root system
of $v^\perp$ is nonempty. In this case we choose a component of
the root system of $v^\perp$ and let $r$ be its highest root. Then the
vector $u=v+r$ has norm $-2(n-1)$, and the assumption about
no norm 0 vectors $z$ with $(z,v)=1$ easily implies that $u$ is still in
the Weyl chamber of $II_{25,1}$. Hence we have reduced $v$ to some known vector
$u$ of norm $-2(n-1)$, and with a little effort it is possible to reverse this
process and construct $v$ from $u$.
\item{3.} Finally suppose that there are no roots in $v^\perp$.
As $v$ is in the Weyl chamber this implies that $(v,r)\le -1$ for all
simple roots $r$. By theorem 1 of chapter 27 there is a norm 0 (Weyl) vector
$w_{25} $ with the property that $(w_{25},r)=-1$ for all simple roots
$r$. Therefore the vector $u=v-w_{25}$ has the property $(u,r)\le 0$ for all
simple roots $r$. So $u$ is in the Weyl chamber, and has norm $-2n -(u,w_{25})$
which is larger than $-2n$ unless $v$ is a multiple of $w_{25}$.
So we can reconstruct $v$ from the known vector $u$ as $v=u+w_{25}$.
In every case we can reconstruct $v$ from known vectors, so we get an algorithm
for classifying the norm $-2n$ vectors in $II_{25,1}$. (This algorithm
breaks down in higher dimensional Lorentzian lattices for two reasons:
it is too difficult to classify the norm 0 vectors, and there is usually
no analogue of the Weyl vector $w_{25}$.)
We now apply the algorithm above to find the 121 orbits of norm $-2$
vectors from the (known) norm 0 vectors, and then apply it again to find the
665 orbits of norm $-4$ vectors from the vectors of norm 0 and $-2$.
The neighbors of a strictly 24 dimensional odd unimodular lattice can
be found as follows. If a norm $-4$ vector $v\in II_{25,1}$
corresponds to the sum of a strictly 24 dimensional odd unimodular
lattice $\Lambda$ and a 1-dimensional lattice, then there are exactly
two norm-0 vectors of $II_{25,1}$ having inner product $-2$ with $v$,
and these norm $0$ vectors correspond to the two even neighbors of
$\Lambda$.
{\bf The enumeration of the odd 24-dimensional lattices.} Figure 17.1
shows the neighborhood graph for the Niemeier lattices, which has a
node for each lattice. If
$A$ and $B$ are neighboring Niemeier lattices, there are three
integral lattices containing $A\cap B$, namely $A$, $B$, and an odd
unimodular lattice $C$ (cf. [Kne4]). An edge is drawn between nodes
$A$ and $B$ in Fig. 17.1 for each strictly 24-dimensional unimodular
lattice arising in this way. Thus there is a one-to-one correspondence
between the strictly 24-dimensional odd unimodular lattices and the
edges of our neighborhood graph. The 156 lattices are shown in Table
17.1. Figure 17.1 also shows the corresponding graphs for dimensions 8
and 16.
For each lattice $\Lambda$ in the table we give its components (in the
notation of the previous chapter) and its even neighbors (represented
by 2 Greek letters as in Table 16.1). The final column gives the
orders $g_1.g_2$ of the groups $G_1(\Lambda)$, $G_2(\Lambda)$ defined
as follows. We may write $Aut(\Lambda)=
G_0(\Lambda).G_1(\Lambda).G_2(\Lambda)$ where $G_0$ is the reflection
group. The group $G_1$ is the subgroup of $Aut(\Lambda)$ of elements
fixing a fundamental chamber of the Weyl group and not interchanging
the 2 neighbors. The group $G_2(\Lambda)$ has order 1 or 2 and
interchanges the two neighbors of $\Lambda$ if it has order 2. (It
turns out that $G_2(\Lambda)$ has order 2 if and only if the two
components of $\Lambda$ are isomorphic.) The components are written as
a union of orbits under $G_1(\Lambda)$, with parentheses around two
orbits if they fuse under $G_2(\Lambda)$.
The first lattice in the table is the odd Leech lattice $O_{24}$,
which is the only one with no norm 2 vectors. The number of norm 2
vectors is given by the formula $$8h(A)+8h(B)-16$$ where $h(A)$ and
$h(B)$ are the Coxeter numbers of the even neighbors of the lattice.
These Coxeter numbers satisfy the inequality $h(B)\le 2h(A)-2$ and the
lattices for which equality holds are indicated by a thick line
in figure 17.1. The Weyl vector $\rho(\Lambda)$ of the lattice $\Lambda$ has
norm given by the
formula $\rho(\Lambda)^2= h(A)h(B)$.
\bye