%This is a plain tex paper.
\magnification=\magstep1
\vbadness=10000
\hbadness=10000
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\def\C{{\bf C}} % complex numbers
\def\Q{{\bf Q}} % rational numbers
\def\R{{\bf R}} % real numbers
\def\Z{{\bf Z}} % integers
\def\Aut{{\rm Aut}}
\proclaim The Leech lattice.
Proc. R. Soc. Lond. A {\bf 398}, 365-376 (1985)
Richard E. Borcherds,
University of Cambridge, Department of Pure Mathematics and
Mathematical Statistics, 16 Mill Lane, Cambridge, CB2 1SB, U.K.
\bigskip
New proofs of several known results about the Leech lattice are
given. In particular I prove its existence and uniqueness and prove
that its covering radius is the square root of 2. I also give a
uniform proof that the 23 ``holy constructions'' of the Leech lattice
all work.
\proclaim 1.~History.
In 1935 Witt found the 8- and 16-dimensional even unimodular lattices
and more than 10 of the 24-dimensional ones. In 1965 Leech found a
24-dimensional one with no roots, called the Leech lattice. Witt's
classification was completed by Niemeier in 1967, who found the twenty
four 24-dimensional even unimodular lattices; these are called the
Niemeier lattices. His proof was simplified by Venkov (1980), who used
modular forms to restrict the possible root systems of such lattices.
When he found his lattice, Leech conjectured that it had covering
radius $\sqrt 2$ because there were several known holes of this
radius. Parker later noticed that the known holes of radius $\sqrt 2$
seemed to correspond to some of the Niemeier lattices, and inspired by
this Conway {\it et al.} (1982a) found all the holes of this
radius. There turned out to be 23 classes of holes, which were
observed to correspond in a natural way with the 23 Niemeier lattices
other than the Leech lattice. Conway (1983) later used the fact that
the Leech lattice had covering radius $\sqrt 2$ to prove that the
26-dimensional even Lorentzian lattice $II_{25,1}$ has a Weyl vector,
and that its Dynkin diagram can be identified with the Leech lattice
$\Lambda$.
So far most of the proofs of these results involved rather long
calculations and many case-by-case discussions. The purpose of this
paper is to provide conceptual proofs of these results. New proofs of
the existence and uniqueness of the Leech lattice and of the fact that it has
covering radius $\sqrt 2$ are given. Finally I prove that the deep
holes of $\Lambda$ correspond to the Niemeier lattices and give a
uniform proof that the 23 ``holy constructions'' of $\Lambda$ found by
Conway \& Sloane (1982b) all work.
The main thing missing from this treatment of the Niemeier lattices is
a simple proof of the classification of the Niemeier lattices. (The
classification is not used in this paper.) By Venkov's results it
would be sufficient to find a uniform proof that there exists a unique
Niemeier lattice for any root system of rank 24, all of whose
components have the same Coxeter number.
Some general results from the papers (Vinberg 1975) and (Venkov 1980)
are quoted; apart from this nearly everything used is proved.
\proclaim 2.~Preliminary results.
We start with some definitions, some results about Niemeier lattices
proved using modular forms, and a description of Vinberg's algorithm.
\proclaim Definitions.
A {\it lattice} $L$ is a finitely generated free $Z$-module with an
integer valued bilinear form, written $(x,y)$ for $x$ and $y$ in
$L$. The {\it type} of a lattice is even (or II) if the {\it norm}
$x^2=(x,x)$ of every element $x$ of $L$ is even, and odd (or I)
otherwise. If $L$ is odd then the vectors in $L$ of even norm form an
even sublattice of index 2 in $L$. $L$ is called {\it positive
definite, Lorentzian, nonsingular,} etc. if the real vector space
$L\otimes \R$ is. ($\R$ stands for the reals and $\Q$ for the
rationals.)
If $L$ is a lattice then $L'$ denotes its {\it dual} in $L\otimes \R$;
i.e. the vectors of $L\otimes \R$ that have integral inner products
with all elements of $L$. $L'$ contains $L$ and if $L$ is nonsingular
then $L'/L$ is a finite abelian group whose order is called the {\it
determinant} of $L$. (If $L$ is singular we say it has determinant 0.)
$L$ is called {\it unimodular} if its determinant is 1.
Even unimodular lattices of a given signature and dimension exist if
and only if there is a real vector space with that signature and
dimension and the signature is divisible by 8. Any two indefinite
unimodular lattices with the same type, dimension, and signature are
isomorphic. $I_{m,n}$ and $II_{m,n}$ $(m\ge 1, n\ge 1)$ are the
unimodular lattices of dimension $m+n$, signature $m-n$, and type $I$
or $II$.
A vector $v$ in a lattice $L$ is called {\it primitive} if $v/n$ is
not in $L$ for any $n>1$. A {\it root} of a lattice $L$ is a norm 2
vector of $L$. Reflection in a root $r$ is then an automorphism of
$L$. This reflection maps $v$ in $L$ to $v-(v,r)r$. A {\it Niemeier
lattice} is an even 24-dimensional unimodular lattice.
The norm 2 vectors in a positive definite lattice $A$ form a root
system which we call the root system of $A$. The hyperplanes
perpendicular to these roots divide $A\otimes \R$ into regions called
{\it Weyl chambers}. The reflections in the roots of $A$ generate a
group called the {\it Weyl group} of $A$, which acts simply
transitively on the Weyl chambers of $A$. Fix one Weyl chamber
$D$. The roots $r_i$ that are perpendicular to the faces of $D$ and
that have inner product at most 0 with the elements of $D$ are called
the simple roots of $D$. (These have opposite sign to what are usually
called the simple roots of $D$. This is caused by the irritating fact
that the usual sign conventions for positive definite lattices are not
compatible with those for Lorentzian lattices. With the convention
used here something is in the Weyl chamber if and only if it has inner
product at most 0 with all simple roots, and a root is simple if and
only if it has inner product at most 0 with all simple roots not equal
to itself.)
The {\it Dynkin diagram} of $D$ is the set of simple roots of $D$. It
is drawn as a graph with one vertex for each simple root of $D$ and
two vertices corresponding to the distinct roots $r,s$ are joined by
$-(r,s)$ lines. (If $A$ is positive definite then two vertices are
always joined by 0 or 1 lines. We will later consider the case that
$A$ is Lorentzian and then its Dynkin diagram may contain multiple
bonds, but these are not the same as the multiple bonds appearing in
$b_n$, $c_n$, $f_4$, and $g_2$.) We use small letters $x_n$ to stand
for spherical Dynkin diagrams. The Dynkin diagram of $A$ is a union of
components of type $a_n$, $d_n$, $e_6$, $e_7$ and $e_8$. The {\it Weyl
vector} $\rho$ of $D$ is the vector in the vector space spanned by
roots of $A$ that has inner product $-1$ with all simple roots of
$D$. It is in the Weyl chamber $D$ and is equal to half the sum of the
positive roots of $D$, where a root is called positive if its inner
product with any element of $D$ is at least 0. The {\it height} of an
element $a$ of $A$ is $-(a,\rho)$, so a root of $A$ is simple if and
only if it has height 1.
The {\it Coxeter number $h$} of $A$ is defined to be the number of
roots of $A$ divided by the dimension of $A$. The Coxeter number of a
component of $A$ is defined as the Coxeter number of the lattice
generated by the roots of that component. Components $a_n$, $d_n$,
$e_6$, $e_7$ and $e_8$ have Coxeter numbers $n+1$, $2n-2$, 12, 18, and
30 respectively. For each component $R$ of the Dynkin diagram of $D$
there is an orbit of roots of $A$ under the Weyl group, and this orbit
has a unique representative $v$ in $D$, which is called the {\it
highest root} of that component. We can write
$-v=n_1r_1+n_2r_2+\cdots$, where the $r$s are the simple roots of $R$
and the $n$s are positive integers called the {\it weights} of the
roots $r_i$. The sum of the $n$s is $h-1$, where $h$ is the Coxeter
number of $R$, and the height of $v$ is $1-h$. The {\it extended
Dynkin diagram} of $A$ is the simple roots of $A$ together with the
highest roots of $A$, and it is the Dynkin diagram of the positive
semidefinite lattice $A\oplus 0$, where $0$ is a one-dimensional
singular lattice. We say that the highest roots of $A$ have weight
one. Any point of weight one of a Dynkin diagram is called a tip of that
Dynkin diagram. We write $X_n$ for the extended Dynkin diagram
corresponding to the Dynkin diagram $x_n$. (Note that $X_{n}$ has
$n+1$ points.)
The automorphism group of $A$ is a split extension of its Weyl group
by $N$, where $N$ is the group of automorphisms of $A$ fixing $D$. $N$
acts on the Dynkin diagram of $D$ and $\Aut(A)$ is determined by its
Dynkin diagram $R$, the group $N$, and the action of $N$ on $R$.
If $A$ is Lorentzian or positive semidefinite then we can still talk
about its root system and $A$ still has a fundamental domain $D$ for
its Weyl group and a set of simple roots. $A$ may or may not have a
Weyl vector but does not have highest roots.
{\it Notation.} Let $L$ be any Niemeier lattice, and let $\Lambda$ be
any Niemeier lattice with no roots. (It will later be proved that
there is a unique such lattice $\Lambda$, called the Leech lattice.)
I give some preliminary results about Niemeier lattices, which are
proved using modular forms. From this viewpoint the reason why
24-dimensional lattices are special is that certain spaces of modular
forms vanish.
\proclaim Lemma 2.1.
(Conway 1969). Every element of $\Lambda$ is congruent $\bmod$
$2\Lambda$ to an element of norm at most 8.
We sketch Conway's proof of this. If $C_i$ is the number of elements
of $\Lambda$ of norm $i$ then elementary geometry shows that the
number of elements of $\Lambda/2\Lambda$ represented by vectors of
norm at most 8 is at least
$$C_0+C_4/2+C_6/2+C_8/(2\times\dim(\Lambda)).$$
The $C_i$'s can be worked out using modular forms and this sum
miraculously turns out to be equal to $2^{24}=$ order of
$\Lambda/2\Lambda$. Hence every element of $\Lambda/2\Lambda$ is
represented by an element of norm at most 8. Q.E.D.
(This is the only numerical calculation that we need.)
Several results from Venkov (1980) are now quoted.
\proclaim Lemma 2.2.
(Venkov 1980) If $y$ is any element of the Niemeier lattice $N$ then
$$\sum(y,r)^2=y^2r^2n/24,$$
where the sum is over the $n$ elements $r$ of some fixed norm.
This is proved using modular forms; the critical fact is that the
space of cusp forms of dimension ${24\over 2}+2=14$ is
zero-dimensional.
\proclaim Lemma 2.3.
(Venkov 1980). The root system of $N$ has rank 0 or 24 and all
components of this root system have the same Coxeter number $h$.
This follows easily from lemma 2.2 as in Venkov (1980). We call $h$
the Coxeter number of the Niemeier lattice $N$. If $N$ has no roots we
put $h=0$.
It is easy to find the 24 root systems satisfying the condition of
lemma 2.3. Venkov gave a simplified proof of Niemeier's result that
there exists a unique Niemeier lattice for each root system. The next four
lemmas are not needed for the proof that $\Lambda$ has covering radius
$\sqrt 2$.
\proclaim Lemma 2.4.
(Venkov 1980). $N$ has $24h$ roots and the norm of its Weyl vector is
$2h(h+1)$.
{\it Proof.} The root system of $N$ consists of components $a_n$,
$d_n$, and $e_n$, all with the same Coxeter number $h$. For each of
these components the number of roots is $hn$ and the norm of its Weyl
vector is ${1\over 12}nh(h+1)$. The lemma now follows because the rank
of $N$ is 0 or 24. Q.E.D.
\proclaim Lemma 2.5.
If $y$ is in $N$ then
$$\sum(y,r)^2=2hy^2,$$
where the sum is over all roots $r$ of $N$.
This follows from lemmas 2.2 and 2.4. Q.E.D.
\proclaim Lemma 2.6.
If $\rho$ is the Weyl vector of $N$ then $\rho$ lies in $N$.
{\it Proof.} We show that if $y$ is in $N$ then $(\rho,y)$ is an
integer, and this will prove that $\rho$ is in $N$ because $N$ is
unimodular. We have
$$\eqalign{ (2\rho,y)^2&= (\sum r,y)^2\qquad\hbox{(The sums are over
all positive roots $r$.)}\cr &=(\sum(r,y))^2\cr
&\equiv\sum(r,y)^2\bmod 2\cr &=y^2h\qquad\hbox{by lemma 2.5}\cr
&\equiv 0\bmod 2 \qquad \hbox{as $y^2$ is even.}\cr }$$ The term
$(2\rho,y)^2$ is an even integer and $(\rho,y)$ is rational, so
$(\rho,y)$ is an integer. Q.E.D.
\proclaim Lemma 2.7.
Suppose $h\ne 0$ and $y$ is in $N$. Then
$$(\rho/h-y)^2\ge 2(1+1/h)$$ and the $y$ for which equality holds form
a complete set of representatives for $N/R$, where $R$ is the
sublattice of $N$ generated by roots.
{\it Proof.} $\rho^2=2h(h+1)$, so
$$\eqalign{
(\rho/h-y)^2-2(1+1/h)
&= [(\rho-hy)^2-\rho^2]/h^2\cr
&=[hy^2-2(\rho,y)]/h\cr
&=[\sum(y,r)^2-\sum(y,r)]/r\quad\hbox{by lemma 2.5}\cr
&= \sum[(y,r)^2-(y,r)]/h,\cr
}$$
where the sums are over all positive roots $r$. This sum is greater
than or equal to 0 because $(y,r)$ is integral, and is zero if and
only if $(y,r)$ is 0 or 1 for all positive roots $r$. In any lattice
$N$ whose roots generate a sublattice $R$, the vectors of $N$ that
have inner product 0 or 1 with all positive roots of $R$ (for some
choice of Weyl chamber of $R$) form a complete set of representatives
for $N/R$, and this proves the last part of the lemma. Q.E.D.
{\it Remark.} This lemma shows that if $N$ has roots then its covering
radius is greater than $\sqrt 2$. (In fact is is at least
$\sqrt{(5/2)}$.) In particular the covering radius of the Niemeier
lattice with root system $A_2^{12}$ is at least $\sqrt{(8/3)}$; this
Niemeier lattice has no deep hole that is half a lattice vector.
I now describe the geometry of Lorentzian lattices and its relation to
hyperbolic space, and give Vinberg's algorithm for finding the
fundamental domains of hyperbolic reflection groups.
Let $L$ be an $(n+1)$-dimensional Lorentzian lattice (so $L$ has
signature $n-1$). Then the vectors of $L$ of zero norm form a double
cone and the vectors of negative norm fall into two components. The
vectors of norm $-1$ in one of these components form a copy of
$n$-dimensional hyperbolic space $H_n$. The group ${\rm Aut}(L)$ is a
product $Z_2\times {\rm Aut}_+(L)$, where $Z_2$ is generated by $-1$
and ${\rm Aut}_+(L)$ is the subgroup of ${\rm Aut}(L)$ fixing each
component of negative norm vectors. See Vinberg (1975) for more
details. (If we wanted to be more intrinsic we could define $H_n$ to
be the set of nonzero negative definite subspaces of $L\otimes \R$.)
If $r$ is any vector of $L$ of positive norm then $r^\perp$ gives a
hyperplane of $H_n$ and reflection in $r^\perp$ is an isometry of
$H_n$. If $r$ has negative norm then $r$ represents a point of $H_n$
and if $r$ is nonzero but has zero norm then it represents an infinite
point of $H_n$.
The group $G$ generated by reflections in roots of $L$ acts as a
discrete reflection group on $H_n$, so we can find a fundamental
domain $D$ for $G$ which is bounded by reflection hyperplanes. The
group ${\rm Aut}_+(L)$ is a split extension of the reflection group by
a group of automorphisms of $D$.
Vinberg (1975) gave an algorithm for finding a fundamental domain $D$
which runs as follows. Choose a vector $w$ in $L$ of norm at most 0;
we call $w$ the {\it controlling vector}. If $r$ is a root then we
will call $-(w,r)/(r,r)$ the {\it height} of $r$. (Roughly speaking,
the height of $r$ measures the distance of $r^\perp$ from $w$.) The
hyperplanes of $G$ passing through $w$ form a root system that is
finite (or affine if $w^2=0$); choose a Weyl chamber $C$ for this root
system. Then there is a unique fundamental domain $D$ of $G$
containing $w$ and contained in $C$, and its simple roots can be found
as follows.
(1) All simple roots have height at least 0.
(2) A root of height 0 is a simple root of $D$ if and only if it is a
simple root of $C$.
(3) A root of positive height is simple if and only if it has inner
product at most 0 with all simple roots of smaller height.
We can now find the simple roots in increasing order of their heights,
and we obtain a finite or countable set of simple roots for $D$. (If
$w^2=0$ there may be an infinite number of simple roots of some
height.) If the hyperplanes of some finite subset of the simple roots
of $D$ bound a region of finite volume, then this subset consists of
all the simple roots of $D$.
\proclaim 3.~Norm zero vectors in Lorentzian lattices.
Here we describe the relation between norm 0 vectors in Lorentzian
lattices $L$ and extended Dynkin diagrams in the Dynkin diagram of
$L$.
{\it Notation.} Let $L$ be any even Lorentzian lattice, so it is
$II_{8n+1,1}$ for some $n$. Let $U$ be $II_{1,1}$; it has a basis of
two norm 0 vectors with inner product $-1$.
If $V$ is any $8n$-dimensional positive even unimodular lattice then
$V\oplus U\cong L$ because both sides are unimodular even Lorentzian
lattices of the same dimension, and conversely if $X$ is any
sublattice of $L$ isomorphic to $U$ then $X^\perp$ is an $8n$-dimensional
even unimodular lattice. If $z$ is any primitive norm 0 vector of $L$
then $z$ is contained in a sublattice of $L$ isomorphic to $U$ and all
such sublattices are conjugate under \Aut$(L)$. This gives 1:1
correspondences between the sets:
(1) $8n$-dimensional even unimodular lattices (up to isomorphism);
(2) orbits of sublattices of $L$ isomorphic to $U$ under \Aut$(L)$; and
(3) orbits of primitive norm 0 vectors of $L$ under \Aut$(L)$.
If $V$ is the $8n$-dimensional unimodular lattice corresponding to the
norm 0 vector $z$ of $L$ then $z^\perp\cong V\oplus 0$ where 0 is the
one-dimensional singular lattice. The Dynkin diagram of $V\oplus 0$ is
the Dynkin diagram of $V$ with all components changed to the
corresponding extended Dynkin diagram.
Now choose coordinates $(v,m,n)$ for $L\cong V\oplus U$, where $v$ is
in $V$, $m$ and $n$ are integers, and $(v,m,n)^2=v^2-2mn$ (so $V$ is
the set of vectors $(v,0,0)$ and $U$ is the set of vectors
$(0,m,n)$). We write $z$ for $(0,0,1)$ so that $z$ is a norm 0 vector
corresponding to the unimodular lattice $V$. We choose a set of simple
roots for $z^\perp\cong V\oplus 0$ to be the vectors $(r_j^i,0,0)$ and
$(r_j^0,0,1)$ where the $r_j^i$'s are the simple roots of the
components $R_j$ of the root system of $V$ and $r_j^0$ is the highest
root of $R_j$. We let $\bar R_j$ be the vectors $(r_j^i,0,0)$ and
$(r_j^0,0,1)$ so that $\bar R_j$ is the extended Dynkin diagram of
$R_j$. Then
$$\sum m_ir_j^i=z\qquad\hbox{and}\qquad\sum m_i=h_j,$$ where the
$m_i$'s are the weights of the vertices of $\bar R_j$ and $h_j$ is the
Coxeter number of $R_j$. We can apply Vinberg's algorithm to find a
fundamental domain of $L$ using $z$ as a controlling vector and this
shows that there is a unique fundamental domain $D$ of $L$ containing
$z$ such that all the vectors of the $\bar R_j$s are simple roots of
$D$.
Conversely suppose that we choose a fundamental domain $D$ of $L$ and
let $\bar R$ be a connected extended Dynkin diagram contained in the
Dynkin diagram of $D$. If we put $z=\sum m_ir_i$, where the $m_i$'s are
the weights of the simple roots $r_i$ of $\bar R$, then $z$ has norm 0
and inner product 0 with all the $r_i$s (because $\bar R$ is an
extended Dynkin diagram) and has inner product at most 0 with all
simple roots of $D$ not in $\bar R$ (because all the $r_i$ do), so $z$
is in $D$ and must be primitive because if $z'$ was a primitive norm 0
vector dividing $z$ we could apply the last paragraph to $z'$ to find
$z'=\sum m_ir_i=z$. The roots of $\bar R$ together with the simple
roots of $D$ not joined to $\bar R$ are the simple roots of $D$
perpendicular to $z$ and so are a union of extended Dynkin
diagrams. This shows that the following 3 sets are in natural 1:1
correspondence:
(1) equivalence classes of extended Dynkin diagrams in the Dynkin
diagram of $D$, where two extended Dynkin diagrams are equivalent if
they are equal or not joined;
(2) maximal disjoint sets of extended Dynkin diagrams in the Dynkin
diagram of $D$, such that no two elements of the set are joined to each
other;
(3) primitive norm 0 vectors of $D$ that have at least one root
perpendicular to them.
\proclaim 4.~Existence of the Leech lattice.
In this section I prove the existence of a Niemeier lattice with no
roots (which is of course the Leech lattice). This is done by showing
that given any Niemeier lattice we can construct another Niemeier
lattice with at most half as many roots. It is a rather silly proof
because it says nothing about the Leech lattice apart from the fact
that it exists.
Leech was the first to construct a Niemeier lattice with no roots
(called the Leech lattice), and Niemeier later proved that it was
unique as part of his enumeration of the Niemeier lattices. In \S 6 I
will give another proof that there is only one such lattice.
{\it Notation.} Fix a Niemeier lattice $N$ with a Weyl vector $\rho$
and Coxeter number $h$, and take coordinates $(y,m,n)$ for
$II_{25,1}=N\oplus U$ with $y$ in $N$, $m$ and $n$ integers.
By lemma 2.6 the vector $w=(\rho,h,h+1)$ is in $II_{25,1}$ and by
lemma 2.4 it has norm 0. We will show that the Niemeier lattice
corresponding to $w$ has at most half as many roots as $N$.
\proclaim Lemma 4.1.
There are not roots of $II_{25,1}$ that are perpendicular to $w$ and
have inner product $0$ or $\pm 1$ with $z=(0,0,1)$.
{\it Proof.} Suppose that $r=(y,0,n)$ is a root that has inner product
0 with $w$ and $z$. Then $y$ is a root of $N$, and $(y,\rho)=nh$
because $(y,\rho)-nh=((y,0,n),(\rho,h,h+1))=(r,w)=0$. But for any root
$y$ of $N$ we have $1\le |(y,\rho)|\le h-1$, so $(y,\rho)$ cannot be a
multiple of $h$.
If $(y,1,n)$ is any root of $II_{25,1}$ that has inner product $-1$
with $z$ and $0$ with $w$ then
$$(y,\rho)-(h+1)-nh=((\rho,h,h+1),(y,1,n))=(w,r)=0$$
and
$$y^2-2n=(y,1,n)^2=r^2=2$$
so
$$\eqalign{
(y-\rho/h)^2&=y^2-2(y,\rho)/h+\rho^2/h^2\cr
&=2+2n-2(nh+h+1)/h+2h(h+1)/h^2\cr
&=2
}$$
which contradicts lemma 2.7. Hence no root in $w^\perp$ can have inner
product 0 or $-1$ with $z$. Q.E.D.
{\it Remark.} In fact there are no roots in $w^\perp$.
\proclaim Lemma 4.2.
The Coxeter number $h'$ of the Niemeier lattice of the vector $w$ is at
most ${1\over 2}h$.
{\it Proof.} We can assume $h'\ne 0$. Let $R$ be any component of the
Dynkin diagram of $w^\perp$ (so $R$ is an extended Dynkin
diagram). The sum $\sum m_ir_i$ is equal to $w$ by \S 3, where
the $r_i$s are the roots of $R$ with weights $m_i$. Also $\sum
m_i=h'$, $(r_i,z)\le -2$ by lemma 4.1, and
$$(\sum m_ir_i,z)=((\rho,h,h+1),z)=-h,$$
so $h'\le {1\over 2} h$. Q.E.D.
\proclaim Theorem 4.3. There exists a Niemeier lattice with no roots.
{\it Proof.} By lemma 4.2 we can find a Niemeier lattice with Coxeter
number at most ${1\over 2}h$ whenever we are given a Niemeier lattice
of Coxeter number $h$. By repeating this we eventually get a Niemeier
lattice with Coxeter number 0, which must have no roots. Q.E.D.
In \S 7 it will be shown that the Niemeier lattice of the vector
$(\rho,h,h+1)$ never has any roots and so is already the Leech
lattice.
\proclaim 5.~The covering radius of the Leech lattice.
Here we prove that any Niemeier lattice $\Lambda$ with no roots has
covering radius $\sqrt 2$.
{\it Notation.} We write $\Lambda$ for any Niemeier lattice with no
roots, and put $II_{25,1}=\Lambda\oplus U$ with coordinates
$(\lambda,m,n)$ with $\lambda$ in $\Lambda$, $m$ and $n$ integers and
$(\lambda,m,n)^2=\lambda^2-2mn$. We let $w$ be the norm 0 vector
$(0,0,1)$ and let $D$ be a fundamental domain of the reflection group
of $II_{25,1}$ containing $w$. If we apply Vinberg's algorithm, using
$w$ as a controlling vector, then the first batch of simple roots to
be accepted is the set of roots $(\lambda, 1, {1\over 2} \lambda^2-1)$
for all $\lambda$ in $\Lambda$. (Conway proved that no other roots are
accepted. See \S 6.) We identify the vectors of $\Lambda$ with
these roots of $II_{25,1} $, so $\Lambda$ becomes a Dynkin diagram
with two points of $\Lambda$ joined by a bond of strength 0,1,2,... if
the norm of their difference is 4,6,8,.... We can then talk about
Dynkin diagrams in $\Lambda$; for example an $a_2$ in $\Lambda$ is two
points of $\Lambda$ whose distance apart is $\sqrt 6$.
Here is the main step in the proof that $\Lambda$ has covering radius
$\sqrt 2$.
\proclaim Lemma 5.1.
If $X$ is any connected extended Dynkin diagram in $\Lambda$ then $X$
together with the points of $\Lambda$ not joined to $X$ contains a
union of extended Dynkin diagrams of total rank 24. (The rank of a
connected extended Dynkin diagram is one less than the number of its
points.)
{\it Proof.} $X$ is an extended Dynkin diagram in the set of simple
roots of $II_{25,1}$ of height 1, and hence determines a norm 0 vector
$z=\sum m_ix_i$ in $D$, where the $x_i$s are the simple roots of $X$
of weights $m_i$. The term $z$ corresponds to some Niemeier lattice
with roots, so by lemma 2.3 the simple roots of $z^\perp$ form a union
of extended Dynkin diagrams of total rank 24, all of whose components
have the same Coxeter number $h=\sum m_i$. Also
$$h=-(z,w)$$ because $z=\sum m_ix_i$ and $(x_i,w)=-1$. By applying
Vinberg's algorithm with $z$ as a controlling vector we see that the
simple roots of $D$ perpendicular to $z$ form a set of simple roots of
$z^\perp$, so the lemma will be proved if we show that all simple
roots of $D$ in $z^\perp$ have height 1, because then all simple roots
of $z^\perp$ lie in $\Lambda$.
Suppose that $Z$ is any component of the Dynkin diagram of $z^\perp$
with vertices $z_i$ and weights $n_i$ so that $\sum n_iz_i=z$. Then
$$-h=(z,w)=(\sum n_iz_i,w)=\sum n_i(z_i,w).$$ Now $(z_i,w)\le -1$ as
there are no simple roots $z_i$ with $(z_i,w)=0$, and $\sum n_i=h$ as
$Z$ has the same Coxeter number $h$ as $X$, so we must have
$(z_i,w)=-1$ for all $i$. Hence all simple roots of $D$ in $z^\perp$
have height 1. Q.E.D.
It follows easily from this that $\Lambda$ has covering radius $\sqrt
2$. For completeness we sketch the remaining steps of the proof of
this. (This part of the proof is taken from Conway {\it et al.}
(1982a).)
{\it Step 1.} The distance between two vertices of any hole is at
most $\sqrt 8$. This follows easily from lemma 2.1.
{\it Step 2.} By a property of extended Dynkin diagrams one of the following must hold for the vertices of any hole.
(i) The vertices form a spherical Dynkin diagram. In this case the
hole has radius $(2-1/\rho^2)^{1\over 2}$, where $\rho^2$ is the norm
of the Weyl vector of this Dynkin diagram.
(ii) The distance between two points is greater than $\sqrt 8$. This
is impossible by step 1.
(iii) The vertices contain an extended Dynkin diagram.
{\it Step 3.} Let $V$ be the set of vertices of any hole of radius
greater than $\sqrt 2$. By step 2, $V$ contains an extended Dynkin
diagram. By lemma 5.1 this extended Dynkin diagram is one of a set of
disjoint extended Dynkin diagrams of $\Lambda$ of total rank 24. These
Dynkin diagrams then form the vertices of a hole $V'$ of radius $\sqrt
2$ whose center is the center of any of the components of its
vertices. Any hole whose vertices contain a component of the vertices
of $V'$ must be equal to $V'$, and in particular $V=V'$ has radius
$\sqrt 2$. Hence $\Lambda$ has covering radius $\sqrt 2$. Q.E.D.
\proclaim 6.~Uniqueness of the Leech lattice.
We continue with the notation of the previous section. We show that
$\Lambda$ is unique and give Conway's calculation of $\Aut(II_{25,1})$.
\proclaim Theorem 6.1.
The simple roots of the fundamental domain $D$ of $II_{25,1}$ are just
the simple roots $(\lambda,1,{1\over 2}\lambda^2-1)$ of height one.
{\it Proof} (Conway 1983). If $r'=(v,m,n)$ is any simple root of
height greater than 1 then it has inner product at most 0 with all
roots of height 1. $\Lambda$ has covering radius $\sqrt 2$ so there is
a vector $\lambda$ of $\Lambda$ with $(\lambda-v/m)^2\le 2$. But then
an easy calculation shows that $(r',r)\ge 1/2m$, where $r$ is the
simple root $(\lambda,1,{1\over 2}\lambda^2-1)$ of height 1, and this
is impossible because $1/2m$ is positive.
\proclaim Corollary 6.2. The Leech lattice is unique, i.e. any two Niemeier
lattices with no roots are isomorphic.
(Niemeier's enumeration of the Niemeier lattices gave the first proof
of this fact. Also see Conway (1969) and Venkov (1980).)
{\it Proof.} Such lattices correspond to orbits of primitive norm 0
vectors of $II_{25,1}$ which have no roots perpendicular to them, so
it is sufficient to show that any two such vectors are conjugate under
$\Aut(II_{25,1})$, and this will follow if we show that $D$ contains
only one such vector. But if $w$ is such a vector in $D$ then theorem
6.1 shows that $w$ has inner product $-1$ with all simple roots of
$D$, so $w$ is unique because these roots generate $II_{25,1}$. Q.E.D.
\proclaim Corollary 6.3.
$\Aut(II_{25,1})$ is a split extension $R.(\cdot\infty)$, where $R$ is
the subgroup generated by reflections and $\pm 1$, and $\cdot \infty$
is the group of automorphisms of the affine Leech lattice.
{\it Proof} (Conway 1983). Theorem 6.1 shows that $R$ is simply
transitive on the primitive norm 0 vectors of $II_{25,1}$ that are not
perpendicular to any roots, and the subgroup of $\Aut(II_{25,1})$
fixing one of these vectors is isomorphic to $\cdot\infty$. Q.E.D.
\proclaim 7.~The deep holes of the Leech lattice.
Conway {\it et al.} (1982a) found the 23 orbits of ``deep holes'' in
$\Lambda$ and observed that they corresponded to the 23 Niemeier
lattices other than $\Lambda$. Conway \& Sloane (1982b,c) later gave
a ``holy construction'' of the Leech lattice for each deep hole, and
asked for a uniform proof that their construction worked. In this
section I give uniform proofs of these two facts.
We continue with the notation of the previous section, so
$II_{25,1}=\Lambda\oplus U$. We write $Z$ for the set of nonzero
isotropic subspaces of $II_{25,1}$, which can be identified with the
set of primitive norm 0 vectors in the positive cone of
$II_{25,1}$. $Z$ can also be thought of as the rational points at
infinity of the hyperbolic space of $II_{25,1}$. We introduce
coordinates for $Z$ by identifying it with the space $\Lambda\otimes
Q\cup \infty$ as follows. Let $z$ be a norm 0 vector of $II_{25,1}$
representing some point of $Z$. If $z=w=(0,0,1)$ we identify it with
$\infty$ in $\Lambda\otimes \Q\cup \infty$. If $z$ is not a multiple of
$w$ then $z=(\lambda,m,n)$ with $m\ne 0$ and we identify $z$ with
$\lambda/m$ in $\Lambda\otimes \Q\cup \infty$. It is easy to check that
this gives a bijection between $Z$ and $\Lambda\otimes \Q\cup \infty$.
The group $\cdot\infty$ acts on $Z$. This action can be described
either as the usual action of $\cdot\infty$ on $\Lambda\otimes \Q\cup
\infty$ (with $\cdot\infty$ fixing $\infty$) or as the action of
$\cdot\infty=\Aut(D)$ on the isotropic subspaces of $II_{25,1}$. We now
describe the action of the whole of $\Aut(II_{25,1})$ on $Z$.
\proclaim Lemma 7.1.
Reflection in the simple root $r=(\lambda,1,{1\over 2} \lambda^2-1)$
of $D$ acts on $\Lambda\otimes \Q\cup \infty$ as inversion in a sphere
of radius $\sqrt 2$ around $\lambda$ (exchanging $\lambda$ and
$\infty$).
{\it Proof.} Let $v$ be a point of $\Lambda\otimes \Q$ corresponding to
the isotropic subspace of $II_{25,1}$ generated by $z=(v,1,{1\over
2}v^2-1)$. The reflection in $r$ maps $z$ to $z-2(z,r)r/r^2$,
$$(z,r)={1\over 2} (z^2+r^2-(z-r)^2)=1-{1\over 2}(v-\lambda)^2,$$
so reflection in $r^\perp$ maps $z$ to
$$\eqalign{ z-(1-{1\over 2}(v-\lambda)^2)r
&=(v,1,{1\over 2}v^2-1)-([1-{1\over 2}(v-\lambda)^2]\lambda,1-{1\over
2}(v-\lambda)^2,?) \cr
&\propto (\lambda+2(v-\lambda)/(v-\lambda)^2,1,?),\cr
}$$
which is a norm 0 vector corresponding to the point
$\lambda+2(v-\lambda)/(v-\lambda)^2$ of $\Lambda\otimes \Q\cup \infty$,
and this is the inversion of $v$ in the sphere of radius $\sqrt 2$
about $\lambda$. Q.E.D.
\proclaim Corollary 7.2.
The Niemeier lattices with roots are in natural bijection with the
orbits of deep holes of $\Lambda$ under $\cdot\infty$. The vertices of
a deep hole form the extended Dynkin diagram of the corresponding
Niemeier lattice.
(This was first proved by Conway {\it et al.} (1982a), who explicitly
calculated all the deep holes of $\Lambda$ and observed that they
corresponded to the Niemeier lattices.)
{\it Proof.} By lemma 7.1 the point $v$ of $\Lambda\otimes \Q$
corresponds to a norm 0 vector in $D$ if and only if it has distance
at least $\sqrt 2$ from every point of $\Lambda$, i.e. if and only if
$v$ is the center of some deep hole of $\Lambda$. In this case the
vertices of the deep hole are the points of $\Lambda$ at distance
$\sqrt 2$ from $v$ and these correspond to the simple roots $(\lambda,
1, {1\over 2} \lambda^2-1)$ of $D$ in $z^\perp$.
Niemeier lattices $N$ with roots correspond to the orbits of primitive
norm 0 vectors in $D$ other than $w$ and hence to deep holes of
$II_{25,1}$. The simple roots in $z^\perp$ form the Dynkin diagram of
$z^\perp\cong N\oplus 0$, which is the extended Dynkin diagram of
$N$. Q.E.D.
Conway \& Sloane (1982b) gave a construction for the Leech lattice
from each Niemeier lattice. They remarked ``The fact that this
construction always gives the Leech lattice still quite astonishes us,
and we have only been able to give a case by case verification as
follows. $\ldots$ We would like to see a more uniform proof''. Here is
such a proof. Let $N$ be a Niemeier lattice with a given set of simple
roots whose Weyl vector is $\rho$. We define the vectors $f_i$ to be
the simple roots of $N$ together with the highest roots of $N$ (so
that the $f_i$s form the extended Dynkin diagram of $N$), and define
the glue vectors $g_i$ to be the vectors $v_i-\rho/h$, where $v_i$ is
any vector of $N$ such that $g_i$ has norm $2(1+1/h)$. By lemma 2.7
the $v_i$s are the vectors of $N$ closest to $\rho/h$ and form a
complete set of coset representatives for $N/R$, where $R$ is the
sublattice of $N$ generated by roots. In particular the number of
$g_i$s is $\sqrt {\det(R)}$.
The ``holy construction'' is as follows.
(i) The vectors $\sum m_if_i+\sum n_ig_i$ with $\sum n_i=0$ form the
lattice $N$.
(ii) The vectors $\sum m_if_i+\sum n_ig_i$ with $\sum m_i+\sum n_i=0$
form a copy of $\Lambda$.
Part (i) follows because the vectors $f_i$ generate $R$ and the
vectors $v_i$ form a complete set of coset representatives for
$N/R$. We now prove (ii).
We will say that two sets are isometric if they are isomorphic as
metric spaces after identifying pairs of points whose distance apart
is 0. For example $N$ is isometric to $N\oplus 0$. Let $z$ be a
primitive norm 0 vector in $D$ corresponding to the Niemeier lattice
$N$. The sets of vectors $f_i$ and $g_i$ are isometric to the sets of
simple roots $f_i'$ and $g_i'$ of $D$, which have inner product 0 or
$-1$ with $z$. (This follows by applying Vinberg's algorithm with $z$
as a controlling vector.)
\proclaim Lemma 7.3.
The vectors $f_i'$ and $g_i'$ generate $II_{25,1}$.
{\it Proof.} The vectors $\sum m_if_i'+n_ig_i'$ with $\sum n_i=0$ are
just the vectors in the lattice generated by $f_i'$ and $g_i'$, which
are in $z^\perp$, and they are isometric to $N$ because the vectors
$\sum m_i f_i+\sum n_ig_i$ with $\sum n_i=0$ form a copy of $N$. The
vector $z$ is in the lattice generated by the $f_i'$s, so the whole of
$z^\perp$ is contained in the lattice generated by the vectors $f_i$
and $g_i$. The vectors $g_i'$ all have inner product 1 with $z$, so
the vectors $f_i'$ and $g_i'$ generate $II_{25,1}$. Q.E.D.
The lattice of vectors $\sum m_if_i'+\sum n_ig_i'$ with $\sum m_i+\sum
n_i=0$ is $w^\perp$ by lemma 2.7 because
$(f_i',w)=(g_i',w)=1$. $w^\perp$ is $\Lambda\oplus 0$, which is
isometric to $\Lambda$ and therefore isomorphic to $\Lambda$
because it is contained in the positive definite space $N\otimes \Q$.
This proves (ii).
The holy construction for $\Lambda$ is equivalent to the
$(\rho,h,h+1)^\perp$ construction of \S 4, so that $(\rho,h,h+1)$
is always a norm 0 vector corresponding to $\Lambda$.
\proclaim References.
Conway, J.H. 1969 A characterisation of Leech's lattice. Invent.
Math. 7, 137-142.
Conway, J.H. 1983 The automorphism group of the
26-dimensional even Lorentzian lattice, J. Algebra 80 (1983) 159-163.
Conway, J.H., Parker, R. A., and Sloane, N.J.A. 1982a The covering
radius of the Leech lattice. Proc. R. Soc. Lond. A 380, 261-290.
Conway, J.H., and Sloane, N.J.A. 1982b Twenty-three constructions for
the Leech lattice. Proc. R. Soc. Lond. A 381, 275-283.
Conway, J.H., and Sloane, N.J.A. 1982c Lorentzian forms for the Leech
lattice. Bull. Am. Math. Soc. 6, 215-217.
Venkov, B.B. 1980 On the classification of integral even unimodular
24-dimensional quadratic forms. Proc. Steklov Inst. Math. 4, 63-74.
Vinberg, \`E. B. 1975 Some arithmetical discrete groups in Loba\v
cevskii space. In discrete subgroups of Lie groups and applications to
moduli, pp. 323-348. Oxford University Press.
\bye