%This is a plain tex paper.
\vbadness=10000
\hbadness=10000
\tolerance=10000
\def\Aut{{\rm Aut}}
\def\Im{{\rm Im}}
\def\Ker{{\rm Ker}}
\def\R{{\bf R}}
\def\Tr{{\rm Tr}}
\def\Z{{\bf Z}}
\proclaim Modular Moonshine II. \hfill 24 July 1994, corrected 19 Sept 1995
Duke Math. J. 83 (1996) no. 2, 435-459.
Richard E. Borcherds,\footnote{$^*$}{ Supported by NSF grant
DMS-9401186.}
Mathematics department,
University of California at Berkeley,
CA 94720-3840
U. S. A.
e-mail: reb@math.berkeley.edu
\smallskip
Alex J. E. Ryba,
Dept of Mathematics,
Marquette University,
Milwaukee,
WI 53233,
U. S. A.
e-mail: alexr@sylow.mscs.mu.edu
\bigskip
The monster simple group acts on the monster vertex algebra, and the
moonshine conjectures state that the traces of elements of the monster
on the vertex algebra are Hauptmoduls. Ryba [R94] conjectured the
existence of similar
vertex algebras over fields of characteristics $p$ acted on by the
centralizers of certain elements of prime order $p$ in the monster,
and conjectured that the Brauer traces of $p$-regular elements of the
centralizers were certain Hauptmoduls. We will prove these
conjectures when the centralizer involves a sporadic group ($p\le 11$,
corresponding to the sporadic groups $B$, $Fi_{24}'$, $Th$, $HN$,
$He$, and $M_{12}$).
\proclaim Contents.
1. Introduction.
Notation.
2. Cohomology and modular representations.
3. Vertex superalgebras mod $p$.
4. A vanishing theorem for cohomology.
5. The case $p=2$.
6. Example: the Held group.
7. Open problems and conjectures.
\proclaim
1.~Introduction.
The original ``moonshine conjectures'' of Conway, Norton, McKay, and
Thompson said that the monster simple group $M$ has an infinite
dimensional graded representation $V=\oplus_{n\in \Z}V_n$ such that
the dimension of $V_n$ is the coefficient of $q^{n}$ of the elliptic
modular function $j(\tau)-744=q^{-1}+196884q+\cdots$, and more
generally the McKay-Thompson series $T_g(\tau)=\sum_{n\in\Z}
\Tr(g|V_n)q^n$ is a Hauptmodul for some genus 0 congruence
subgroup of $SL_2(\R)$. The representation $V$ was constructed by
Frenkel, Lepowsky, and Meurman [FLM], and it was shown to satisfy the
moonshine conjectures in [B92] by using the fact that it carries the
structure of a vertex algebra [B86], [FLM].
Meanwhile, Norton had suggested that there should be a graded space
associated to every element $g$ of the monster acted on by some
central extension of the centralizer of $g$ ([N], see also [Q]). It is
easy to see that these graded spaces are usually unlikely to have
a vertex algebra structure. Ryba suggested [R94] that these spaces
might have a vertex algebra structure if they were reduced mod $p$ (if
$g$ has prime order $p$). He also suggested the following definition
for these vertex algebras:
$$^gV= {V^g/pV^g\over (V^g/pV^g)\cap(V^g/pV^g)^\perp}$$
where $V$ is some integral form for the monster
vertex algebra, and $V^g$ is the set of vectors fixed by $g$. He
conjectured that the dimensions of the homogeneous components of $^gV$
should be the coefficients of certain Hauptmoduls, and more generally
that if $h$ is a $p$-regular element of $C_M(g)$ then $$
\Tr(h|^gV_n)=\Tr(gh|V_n)$$ where the trace on the left is the Brauer
trace. (The numbers on the right are known to be the coefficients of a
Hauptmodul depending on $gh$ by [B92].) We will call these conjectures
the modular moonshine conjectures.
We now describe the proof of the modular moonshine conjectures for
some elements $g$ given
in this paper. We observe (proposition 2.3)
that the definition of $^gV$ in
[R94] is equivalent to defining it to be the Tate cohomology group
$\hat H^0(g,V)$, or at least it would be if a good integral form $V$
of the monster Lie algebra was known to exist. The lack of a good
integral form $V$ does not really matter much, because we can just as
well use a $\Z[1/n]$-form for any $n$ coprime to $|g|$, and a $\Z[1/2]$-form
can be extracted from Frenkel, Lepowsky and Meurman's
construction of the monster vertex algebra.
It is difficult to work out the dimension of $\hat H^0(g,V[1/2]_n)$
directly. However it is easy to work out the difference of the
dimensions of $\hat H^0$ and $\hat H^1$, which can be thought of as a
sort of Euler characteristic (and is closely related to the
Herbrand quotient in number theory). This suggests that we should
really be looking at $\hat H^*(g,V[1/2])=\hat H^0(g,V[1/2])\oplus \hat
H^1(g,V[1/2])$, and it turns out for essentially formal reasons that
this has the structure of a vertex superalgebra. The traces of elements
$h$ of the centralizer of $g$ on this superalgebra are just given by
the traces of the elements $gh$ on the monster vertex algebra.
We can now complete the proof of the modular moonshine conjectures for
the element $g$ of the monster by showing that $\hat H^1(g,V[1/2])=0$.
We do this for certain elements of odd prime order of the monster
coming from $M_{24} $ in section 4; this is a long but straightforward
calculation. There are plenty of elements $g$ of the monster for
which $\hat H^1(g,V[1/2])$ does not vanish; this happens whenever some
coefficient of the Hauptmodul of $g$ is negative; for example, $g$ of
type 3B. There are also some elements of large prime order for which
we have been unable to prove the modular moonshine conjectures because
we have not proved that $\hat H^1(g,V[1/2])=0$ (though this is probably
true), but we do at least cover all the cases when $g$ has prime order
and its centralizer involves a sporadic simple group.
The case when $p=2$ has several extra complications, due partly to the
existence of several extensions of groups by 2-groups, and due partly
to the fact the FLM construction can be carried out over $\Z[1/2]$
but it is not clear how to do it over $\Z$. We deal with these extra
problems in section 5 by using a $\Z[1/3]$-form of the monster vertex
algebra. To construct this $\Z[1/3]$-form we need to make a mild
assumption (which we have not checked) about the
construction of the monster vertex algebra from an element of order 3
announced by Dong and Mason and by Montague. (In particular the proof
of the modular moonshine conjectures for elements of type 2A in the
monster uses this assumption.)
In section 6 we give some calculations illustrating the case when $g$
is an element of type 7A in the monster (so the centralizer $C_M(g)$
of $g$ is $\langle g
\rangle \times He$). We give the characters and the decomposition into
irreducible modular representations of the first few graded pieces of
$\hat H^0(g,V[1/2])$.
In section 7 we discuss some open problems, in particular how one
might construct a good integral form of the monster vertex algebra.
\proclaim
Notation.
\item{$A^G$} The largest submodule of $A$ on which $G$ acts trivially.
\item{$A_G$} The largest quotient module of $A$ on which $G$ acts trivially.
\item{$A,B,C$} $G$-modules.
\item{$\Aut$} The automorphism group of something.
\item{$B$} The baby monster sporadic simple group.
\item{${\bf C}$} The complex numbers.
\item{$C_M(g)$} The centralizer of $g$ in the group $M$.
\item{$Fi_{24}$} One of Fischer's groups.
\item{$\bf F_q$} The finite field with $q$ elements.
\item {$g$} An element of $G$, usually of order $p$.
\item{$\langle g\rangle$} The group generated by $g$.
\item{$G$} A group, often cyclic of prime order $p$ and generated by $g$.
\item{$\hat H^i(G,A)$} A Tate cohomology group of the finite group $G$
with coefficients in the $G$-module $A$.
\item{$\hat H^i(g,A)$} means $\hat H^i(\langle g\rangle,A)$, where $\langle g\rangle$
is the cyclic group generated by $g$.
\item{$\hat H^*(g,A)$} The sum of the Tate cohomology groups
$\hat H^0(g,A)$ and $\hat H^1(g,A)$, considered as a super module.
\item{$He$} The Held sporadic simple group.
\item{$HN$} The Harada-Norton sporadic simple group.
\item{$I$} The elements of a group ring of norm 0.
\item {$\Im$} The image of a map.
\item{$\Ker$} The kernel of a map.
\item{$\Lambda,\hat\Lambda$} The Leech lattice and a double cover of the Leech
lattice.
\item{$L$} An even lattice.
\item{$M$} The monster simple group.
\item{$M_{12}$} A Mathieu group.
\item{$M_{24}$} A Mathieu group.
\item{$N$} The norm map: $N(v)=\sum_{g\in G}g(v)$.
\item{$N_M(g)$} The normalizer of the subgroup $\langle g\rangle$ in the group $M$.
\item{$p$} A prime, usually the order of $g$.
\item{$\R $} The real numbers.
\item{$R_p$} A finite extension of the $p$-adic integers.
\item{$S(a)$}$=\sum_{0\le n 3$, and
any element of $V$ is a sum of eigenvectors of
the operator $L_0=\omega_{1}$ with integral eigenvalues.
If $v$ is an eigenvector of $L_0$, then its eigenvalue is called
the (conformal) weight of $v$. If $v$ is an element of the monster
vertex algebra $V$ of conformal weight $n$, we say that
$v$ has degree $n-1$ ($=n-c/24$), and we write $V_n$ for the
module of elements of $V$ of degree $n$.
The $\R$-form of the vertex algebra of any $c$-dimensional even lattice $L$
has a canonical
conformal vector $\omega=\sum_{i}a_i(1)a_i(1)/2$ of dimension $c$,
where the elements $a_i$ run over an orthonormal basis
of $L\otimes \R$ and the monster vertex algebra
has a conformal vector of dimension 24.
If $\omega$ is a conformal vector of a vertex algebra $V$ then we define
operators $L_i$ on $V$ for $i\in \Z$ by
$$L_i = \omega_{i+1}.$$
These operators satisfy the relations
$$[L_i,L_j] = (i-j)L_{i+j} + {i+1\choose 3}{c\over 2}\delta^i_{-j}$$
and so make $V$ into a module over the Virasoro algebra.
The operator $L_{-1}$ is equal to $D$.
The vertex algebra of any even lattice $L$
has a real valued symmetric bilinear form (,) such that
the adjoint of the operator $u_n$ is
$(-1)^i\sum_{j\geq 0}L_1^j(\sigma(u))_{2i-j-n-2}/j!$ if $u$ has degree $i$,
where $\sigma$ is the automorphism of the vertex algebra
defined by $\sigma(e^w)=(-1)^{(w,w)/2}(e^w)^{-1}$ for $e^w$
an element of the twisted group ring of $L$ corresponding to the
vector $w\in L$.
Similarly the monster vertex algebra
has a real valued symmetric bilinear form (,) such that
the adjoint of the operator $u_n$ is
$(-1)^i\sum_{j\geq 0}L_1^j(u)_{2i-j-n-2}/j!$ if $u$ has degree $i$.
If a vertex
algebra has a bilinear form with the properties above we say that the
bilinear form is
compatible with the conformal vector. If a vertex algebra does
not have a conformal vector but only a $\Z$-grading we can still
define compatible bilinear forms, because we can define the
operator $L_1^j/j!$ to be the adjoint of the operator
$L_{-1}^i/i!=D^{(i)}$. (If we are not in characteristic 0 we have to
modify these definitions slightly by replacing $L_1^i/i!$ by
a system of divided powers of $L_1$ satisfying some conditions.)
The integral form $V_\Lambda$ of the vertex algebra of the
Leech lattice contains the conformal vector
$\omega =\sum_{1\le i\le 24}a_i(1)a_i(1)/2$ where the $a_i$'s run over
an orthogonal basis of $\Lambda\otimes \R $ because we can rewrite
this as $\omega = \sum_{1\le i3$ or $p\le 3$. If $p>3$ then $\omega$ has norm $12 \bmod
p$, which is nonzero so that $\hat H^0(g,V[1/2]_1)$ splits as the
orthogonal direct sum of a 1-dimensional space spanned by $\omega$ and
its orthogonal complement (and this splitting is obviously invariant
under $C_M(g)$). If $p=3$ and $g$ is an element of type $3A$ then
$\omega$ has norm $12\equiv 0 \bmod 3$ so it is contained in its
orthogonal complement, and $\hat H^0(g,V[1/2]_1)$ has a composition
series of the form $1.781.1$ (as a $C_M(g)/\langle g\rangle=Fi_{24}'$
module). The Atlas [C] states that the 781 dimensional module has an
algebra structure mod 3, but this seems to be a mistake and the
construction there only gives an algebra structure on 1.781.1 mod 3.
When $g$ is in the class 3C then the vertex algebra
$\hat H^0(g,V[1/2])$
does not have a conformal vector, and in fact not only
is the image of $\omega$ equal to zero, but the whole degree 1 space
$\hat H^0(g,V[1/2]_1)$ is zero. However we can turn $\hat
H^0(g,V[1/2])$ into a better vertex algebra by ``compressing'' it.
If we look at the series $q^{-1}+248q^2+4124q^5+\cdots$ we see that
only every 3rd term of the graded space $\hat H^0(g,V[1/2])$ is
nonzero. Hence we change the grading of the piece of degree $3n-1$ to
$n$. We also change the vertex operator $v(x)$ of $v\in \hat
H^0(g,V[1/2])$ to $v(x^{1/3})$. It is easy to check that this
defines a new vertex algebra structure on $\hat H^0(g,V[1/2])$,
(because if we are in characteristic 3 then $x-y=(x^{1/3}-y^{1/3})^3$),
whose homogeneous degrees are the coefficients of
$1+248q+4124q^2+\cdots$. The compression of the
vertex algebra $\hat H^0(g,V[1/2])$
is probably isomorphic to the vertex
algebra $V_{E_8}/3V_{E_8}$ (the vertex algebra of the $E_8$ lattice
reduced mod 3) which is acted on by the finite group $E_8({\bf F_3})$,
because both vertex algebras have the same graded dimension
and are both acted on by the Thompson group.
(It is well known that the Thompson group $C_M(g)/\langle g\rangle$ is
contained in $E_8({\bf F_3})$.) Warning: the vertex algebra
$V_{E_8}/3V_{E_8}$ can be lifted to characteristic 0, but the the
action of the Thompson group does not lift to an action on the vertex
algebra $V_{E_8}$.
\proclaim
5.~The case $p=2$.
In this section we will extend the proof of the modular moonshine
conjectures (Corollary 4.8) to cover the case of the elements of type
2A in the monster, to obtain a vertex algebra over ${\bf F_2}$ acted
on by the baby monster. The proof is similar to the cases for elements
of odd order, except that there are several extra technical
complications, and we have to use one assertion which we have not
completely proved.
The theorems in this section depend on the following assumption about
the monster vertex algebra. We have not rigorously verified
this, but we sketch how its verification should go.
\proclaim Assumption 5.1. There is a $\Z[1/3]$-form $V[1/3]$
of the monster vertex algebra with a self dual bilinear form such that
$V[1/3]$ can be written as a direct sum of $2.M_{12}.2$
modules each of which is isomorphic to a submodule of the $\Z[1/3]$-form
$V_\Lambda[1/3]$ of $V_{\Lambda}$.
We outline a possible proof of 5.1. (Roughly speaking, this outline
would become a correct proof if Dong, Mason or Montague do not need to
divide by any integer other than 3 in their constructions.) We
obviously cannot use the FLM construction of the monster vertex
algebra, which writes $V$ as the sum of the eigenspaces of an element
of type 2B, because this involves inverting 2 and gives a 2-divisible
module whose cohomology vanishes. Instead we use the construction of
the monster vertex algebra as a sum of eigenspaces of an element of
type 3B, due independently to Dong and Mason [DM] and Montague [M]. This
should produce a $\Z[1/3,\omega]$-form $V[1/3,\omega]$ of the monster
vertex algebra as follows, where $\omega$ is a cube root of 1
satisfying $\omega^2+\omega+1=0$. We first note that
modules over $\Z[1/3,\omega]$ can be written as the sums of eigenspaces
of any group $\langle h\rangle$ of order 3 acting on them,
because $a={a+h(a) +h^2(a)\over 3}+{a+\omega h(a) +\omega^2h^2(a)\over 3}+
{a+\omega^2h(a) +\omega h^2(a)\over 3}$.
The vertex algebra $V[1/3,\omega]$
is a sum of 3 submodules $V^0, V^1$, and $V^2$, which are the
eigenspaces of an element $h$ of type 3B. The space $V^0$ is
isomorphic to the subspace of $V_\Lambda[1/3,\omega]$ fixed by an
element of order 3 coming from a fixed point free element of order 3
of $\Aut(\Lambda)$, and this defines the $\Z[1/3,\omega]$-form of
$V^0$. The spaces $V^i$ each split into the sum of 3 subspaces
$V^{ij}$ which are the eigenspaces of a group $(\Z/3\Z)^2$ in the
monster, and Dong and Mason and Montague show that there is a group $SL_2({\bf
F_3})$ acting transitively on the spaces $V^{ij}$ for $(i,j)\ne
(0,0)$. We use this group to transport the $\Z[1/3,\omega]$-form from
$V^{01}$ and $V^{02}$ to $V^{ij}$ for $i\ne 0$. This defines the
$\Z[1/3,\omega]$-form on the monster vertex algebra (and it also has a
self dual $\Z[1/3,\omega]$ bilinear form and a conformal vector,
coming from the same structures on $V_{\Lambda}$).
We now reduce this $\Z[1/3,\omega]$-form to a $\Z[1/3]$-form. We have
an operation of complex conjugation on $V^0$ coming from complex
conjugation on $V_\Lambda\otimes \Z[1/3,\omega]$, and the action of
$SL_2({\bf F_3})$ can be used to transfer this to complex conjugation
on all the spaces $V^{ij}\oplus V^{2i,2j}$, and hence to
$V[1/3,\omega]$. This conjugation preserves the vertex algebra
structure and commutes with the monster, and satisfies
$\overline{\omega v}=\bar\omega\bar v$.
Any element of
$V[1/3,\omega]$ can be written uniquely in the form $a+\omega b$ with
$a$ and $b$ fixed by conjugation, because finding $a$ and $b$ involves
solving the 2 linear equations $v=a+b\omega$, $\bar v = a +\bar\omega
v$, and there is a unique solution for $a$ and $b$ because the
determinant of this system of equations is $\omega-\bar\omega$ which
is a unit in $\Z[1/3,\omega]$. Hence if we define $V[1/3]$ to be the
fixed points of the conjugation, we see that
$V[1/3]\otimes\Z[1/3,\omega]=V[1/3,\omega]$.
Hence $V[1/3] $ is a
$\Z[1/3]$-form for the monster vertex algebra. This completes the
arguments for assumption 5.1.
\proclaim Theorem 5.2. If we assume that 5.1 is correct then
there is a vertex algebra defined over ${\bf F_2}$ acted on by the
baby monster, such that the McKay-Thompson series of every element of
the baby monster is a Hauptmodul. The vertex algebra has a conformal
vector and a compatible self dual bilinear form, also invariant under
the baby monster.
Proof. We will only give details for parts of the argument that differ
significantly from the proof of corollary 4.8. We choose an element
$g\in M_{24}\subset\Aut(\Lambda)$ of order 2 and trace 8. (There is a
unique conjugacy class of such elements in $M_{24}$.)
The first step is to prove that $\hat H^1( g,
\Lambda)=\Lambda_{\langle g\rangle}/\Ker(N)=0$, which we do by direct
calculation as follows. We can assume that $g$ acts on $\Lambda$ in
the usual coordinates by acting as $-1$ on the first 8 coordinates and
as $+1$ on the last 16. Then $\Ker(N)$ is the sublattice of $\Lambda$
of vectors whose last 16 coordinates vanish. This lattice is a copy
of the $E_8$ lattice with norms doubled (so it has 240 vectors of norm
4 and so on). It is generated by its norm 4 vectors and the
centralizer of $g$ acts transitively on these norm 4 vectors, so to
prove that $\Lambda_{\langle g\rangle}=\Ker(N)$ we only need to prove
that one of these norm 4 vectors is of the form $v-g(v)$ for some
vector $v\in \Lambda$. We can do this by taking $v$ to be the vector
$(1^{23}, -3)$.
Therefore $\hat H^1( g, \Lambda)=0$.
From this it follows as in theorem 4.6 that $\hat H^1( \hat g,
V_\Lambda)=0$, provided we show that $g$ lifts to an element in
$\Aut(\hat\Lambda)$ which we denote by $\hat g$, such that $\hat g$
also has order 2 and $\hat g$ fixes every element $e^a$ of the twisted
group ring of $\Lambda$ corresponding to an element $a$ of $\Lambda$
fixed by $g$. (If $g$ is of odd order the existence of a good element
$\hat g$ is trivial, but for some elements of even order, for example
those of trace 0, a lift with these properties does not exist.) Lemma
12.1 of [B92] says that such a lift $\hat g$ exists provided that
$(v,g(v))$ is even for all elements $v\in \Lambda$, which follows
because $(v,g(v))= ((v-g(v),v-g(v))-(v,v)-(g(v),g(v)))/2$, and
$(v,v)=(g(v),g(v))$ is even while $v-g(v)\in
\Ker(N)$ has norm divisible by 4.
Hence $\hat H^1( \hat g, V_\Lambda)=0$.
In particular any submodule of $V_{\Lambda}[1/3]$ invariant under
$\hat g$ also has vanishing $\hat H^1$, so by assumption 5.1 $\hat
H^1(\hat g,V[1/3])=0$. We can now follow the argument proving
corollary 4.8 to construct a vertex algebra satisfying the conditions
of theorem 5.2. This proves theorem 5.2.
Remark. In the proof of theorem 5.2 we should really have checked that
the 2 vertex algebras constructed by Frenkel-Lepowsky-Meurman and by
Dong-Mason-Montague are
isomorphic as modules over the monster, so that the traces of elements
of the monster are Hauptmoduls. Dong and Mason have announced [DM]
that these two vertex algebras are indeed isomorphic as vertex
algebras acted on by the monster. It would also be easy to show that
they are isomorphic as modules over the monster (but not
as vertex algebras) by using the results of [B92] to calculate
the character of the Dong-Mason-Montague algebra.
It seems likely that $V[1/2]$ and $V[1/3]$ both come from a
conjectural integral form $V$ of the monster vertex algebra by
tensoring with $\Z[1/2]$ or $\Z[1/3]$; see lemma 7.1.
If we look at the McKay-Thompson series for the element 2B of the
monster we see that the coefficients are alternating, which suggests
that $\hat H^0$ vanishes for half the homogeneous pieces of $V$, and
$\hat H^1$ vanishes for the other half. The next theorem shows that
this is indeed the case.
\proclaim Theorem 5.3. Assume that 5.1 is correct.
If $\hat g$ is an element of type 2B in the monster, then $\hat
H^0(\hat g,V[1/3]_n)$ vanishes if $n$ is even, and $\hat H^1(\hat
g,V[1/3]_n)$ vanishes if $n$ is odd.
Proof. By using the argument in the last few paragraphs of
the proof of theorem 5.2
we see that it is sufficient to prove the same result for the
cohomology with coefficients in $V_\Lambda$ for a suitable element
$\hat g$ of $\Aut(\hat\Lambda)$. We will take $g$ to be the
automorphism $-1$ of $\Lambda$, and take its lift $\hat g$ in
$\Aut(\hat\Lambda)$ to be the element taking $e^a$ to
$(-1)^{(a,a)/2}(e^a)^{-1}$ in the group ring. It is easy to check that
$\hat g$ is an automorphism of order 2. As a graded $\hat g$ module,
$V_\Lambda$ is the tensor product of the twisted group ring and the
module $V_{\Lambda,0}$ as in theorem 4.6. The twisted
group ring is a sum of infinitely many free modules over the group
ring and one copy of $\Z$ (spanned by 1), so its 0'th cohomology group
is $\Z/2\Z$, and its first cohomology vanishes. Hence
$H^*(\hat g, V_\Lambda)=H^*(\hat g, V_{\Lambda,0})$.
We let $\omega$ be the involution of $V_{\Lambda,0}$ that is the product of
$\hat g$ and the involution acting as $-1^n$ on the piece of degree $n$.
As multiplying the action of $\hat g$ by $-1$ exchanges $H^0$ and $H^1$,
we see that to prove theorem 5.3 we have to show that
$H^1(\omega, V_{\Lambda,0})=0$. To do this
it is sufficient to show that $V_{\Lambda,0}$
is a permutation module for $\omega$.
We now complete the proof of theorem 5.3 by finding an explicit
basis of $V_{\Lambda,0} $ described in terms of S-functions, on which
$\omega$ acts as a permutation.
If we choose a basis $a_1,\ldots, a_{24}$
of $\Lambda$, then $V_{\Lambda,0}$ is the polynomial ring
generated by the elements $h_{i,n}=e^{-a_i}D^{(n)}e^{a_i}$, $1\le i\le 24$,
$n\ge 1$. The element $\hat g$ takes $h_{i,n}$ to
$e^{a_i}D^{(n)}e^{-a_i}$, so $\omega(h_{i,n})=e_{i,n}$, where
$e_{i,n}= (-1)^ne^{a_i}D^{(n)}e^{-a_i}$. From the relation
$$\sum_mD^{(m)}(e^{a_i})D^{(n-m)}(e^{-a_i})=D^{(n)}(e^{a_i}e^{-a_i})=0$$
for $n\ge 1$ we find that
$$\sum_{0\le m\le n}(-1)^me_{i,m}h_{i,n-m}=0$$
for $n\ge 1$, which recursively defines the $e_{i,n}$'s in terms of
the $h_{i,n}$'s. If we identify the $h_{i,n}$'s with the $n$'th complete
symmetric function (see [Mac], page 14) then the ring generated by the
$h_{i,n}$'s for some fixed $i$ is the ring of all symmetric functions.
By the formulas 2.6' and 2.7 on page 14 of [Mac] we see that
the $e_{i,n}$'s are then identified with the elementary symmetric functions,
and $\omega$ with the canonical involution $\omega$ of [Mac].
The ring of all symmetric functions is spanned by the S-functions
$s_\lambda$ for permutations $\lambda$ [Mac, p. 24, formulas 3.1 and 3.3],
and the action of $\omega$ on the S-functions is given by
$$\omega(s_\lambda)=s_{\lambda'}$$
[Mac, p. 28, formula 3.8] where $\lambda'$ is the conjugate partition
of $\lambda$ [Mac, p. 2]. Therefore the ring generated by the
$h_{i,n}$'s for any fixed $i$ is a permutation module for $\omega$,
and therefore so is $V_{\Lambda,0}$ because it is the tensor product
of 24 permutation modules.
This proves theorem 5.3.
Of course, as we are working in characteristic 2, there is really no
difference between a vertex algebra and a vertex superalgebra, so the
superalgebra associated to 2B can be considered as a vertex algebra
with dimensions given by the coefficients of the series
$q^{-1}+276q+2048q^2+\cdots$.
However it seems more natural to think
of it as a superalgebra.
This superalgebra is acted on by the group $2^{24}.Co_1$. The next
paragraph suggests that the $2^{24}$ may act trivially, so that we get
an action of $Co_1$, but we have not proved this.
We can also ask if this superalgebra can be lifted to characteristic
0. It turns out that there is a vertex superalgebra (constructed
below) with the right grading acted on by $\Aut(\Lambda)$. In fact
this superalgebra has 2 different natural actions of $\Aut(\Lambda)$,
one faithful, and one for which the center of $\Aut(\Lambda)$ acts
trivially. Unfortunately we do not know of an integral form invariant
under either action of $\Aut(\Lambda)$, but it seems reasonable to
guess that both actions have invariant integral forms, and the
reduction mod 2 of either integral form is the mod 2 vertex
superalgebra above. (This requires that the $2^{24}$ acts trivially
on this mod 2 vertex algebra.)
We can construct this superalgebra in characteristic 0 easily as
the vertex superalgebra of the odd 12-dimensional unimodular lattice
with no roots $D_{12}$, which is the set of vectors $(x_1,\ldots,
x_{24})$ in $\R ^{24}$ such that all the $x_i$'s are integers or
all are integers $+1/2$, and their sum is even. This vertex
superalgebra is acted on by the spinor group $Spin_{12}(\R)$. The
group $\Aut(\Lambda)$ in $SO_{24}(\R)$ can be lifted to the spin
group as it has vanishing Schur multiplier, so we get an action of
$\Aut(\Lambda)$ on the vertex algebra. The reason why we get two
actions is that there are two conjugacy classes of embeddings of
$\Aut(\Lambda)$ in $SO_{24}(\R)$ (which are interchanged by a
reflection). The spin group does not act faithfully because an
element of order 2 in the center acts trivially, and it is not hard to
check that for exactly one of the classes of embeddings of
$\Aut(\Lambda)$ in $SO_{24}(\R)$ the element $-1\in
\Aut(\Lambda)$ lifts to the element of $Spin_{24}(\R)$ acting
trivially.
The existence of these two actions of $\Aut(\Lambda)$ has the curious
consequence that every second coefficient of
$q^{-1}+276q+2048q^2+\cdots$ is in a natural way the dimension of 2
different representations of $\Aut(\Lambda)$, in which the nontrivial
element of the center acts as either $+1$ or $-1$. For example 2048
decomposes as either $1771+276+1$ or $2024+24$.
\proclaim
6.~Example: the Held group.
We give some numerical tables to illustrate the case when $g$ is an
element of type 7A in the monster, with centralizer $\langle
g\rangle\times He$.
First we give the 7-modular character table of
$He$ (taken from [R88]), followed by the modular characters of the
first few head characters of $\hat H^0(g,V[1/2])$ (which can be read
off from [CN] table 4). The bottom line gives the corresponding
conjugacy classes in the monster group (the one whose Hauptmodul is
given by the coefficients of the head characters).
\vfill \line{}
\vbox{
\halign{$ #$\hfil &&\ \hfil$ #$\cr
& 1A& 2A& 2B& 3A&
3B&4A&4B&4C&5A&6A&6B&8A&10A&12A&12B&15A&17A&17B\cr & 1& 1& 1& 1& 1& 1&
1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\cr & 50& 10& 2& 5&-1& 2& 2&-2& 0&
1&-1& 0& 0&-1&-1& 0&-1&-1\cr & 153& 9& -7& 0& 3&-3& 1& 1& 3& 0&-1&
1&-1& 0& 1& 0& 0& 0\cr & 426& 26& 10& 3&-3&-2& 2& 2& 1&-1& 1&-2& 1&
1&-1&-2& 1& 1\cr & 798& 38& 14& 6& 3& 6&-2& 2&-2& 2&-1& 0&-2& 0& 1&
1&-1&-1\cr & 1029& -35& 21& 21& 0&-7&-3& 1& 4& 1& 0&-1& 0&-1& 0& 1&
{1-\sqrt{17}\over 2}&{1+\sqrt{17}\over 2}\cr & 1029& -35& 21& 21&
0&-7&-3& 1& 4& 1& 0&-1& 0&-1& 0& 1& {1+\sqrt{17}\over 2}&
{1-\sqrt{17}\over 2}\cr & 1072& 16&-16& 10&-2& 0& 0& 0&-3&-2& 2& 0& 1&
0& 0& 0& 1& 1\cr & 1700& 20& 4&-10& 5& 0&-4&-4& 0& 2& 1& 0& 0& 0&-1&
0& 0& 0\cr & 3654& -10& 38& -9& 3& 2&-2&-2& 4&-1&-1& 2& 0&-1& 1&
1&-1&-1\cr & 4249& 9& -7& -8&-5&-3& 1& 1&-1& 0&-1& 1&-1& 0& 1&
2&-1&-1\cr & 6154& -70&-22& -2& 7& 6& 2& 2& 4& 2&-1&-2& 0& 0&-1&-2& 0&
0\cr & 6272& -64& 0& 35& 8&-8& 0& 0&-3&-1& 0& 0& 1& 1& 0& 0&-1&-1\cr &
7497& 81& -7& 0& 3&-3& 1&-3&-3& 0&-1&-1& 1& 0& 1& 0& 0& 0\cr & 13720&
-56& 56&-14& 7& 0& 8& 0&-5&-2&-1& 0&-1& 0&-1& 1& 1& 1\cr & 14553& 9&
-7& 0&-9& 9&-7& 1& 3& 0&-1& 1&-1& 0&-1& 0& 1& 1\cr & 17493& 21&
21&-21& 0&-7&-3& 5&-7& 3& 0& 1& 1&-1& 0&-1& 0& 0\cr & 23324&-196& 28&
14&-7& 0& 4&-4&-1& 2& 1& 0&-1& 0& 1&-1& 0& 0\cr
\cr
H_{-1}& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\cr H_{0}
& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr H_{1} & 51&
11& 3& 6& 0& 3& 3&-1& 1& 2& 0& 1& 1& 0& 0& 1& 0& 0\cr H_{2} & 204& 20&
-4& 6& 3& 0& 4& 0& 4& 2&-1& 2& 0& 0& 1& 1& 0& 0\cr H_{3} & 681& 57& 9&
15& 0& 1& 9& 1& 6& 3& 0& 1& 2& 1& 0& 0& 1& 1\cr H_{4} & 1956& 92&-12&
30& 0& 0&12& 0& 6& 2& 0& 2& 2& 0& 0& 0& 1& 1\cr H_{5} & 5135& 207& 15&
41& 8& 7&15&-1&10& 9& 0& 3& 2& 1& 0& 1& 1& 1\cr H_{6} & 12360&
312&-24& 66& 0& 0&24& 0&10& 6& 0& 4& 2& 0& 0& 1& 1& 1\cr H_{7} &
28119& 623& 39&111& 0& 7&39& 3&19&11& 0& 5& 3& 1& 0& 1& 1& 1\cr H_{8}
& 60572& 932&-52&146&11& 0&52& 0&22&14&-1& 6& 2& 0& 1& 1& 1& 1\cr
H_{9} &125682&1674& 66&222& 0&18&66&-2&32&18& 0& 8& 4& 0& 0& 2& 1&
1\cr H_{10}&251040&2464&-96&336& 0& 0&96& 0&40&16& 0& 8& 4& 0& 0& 1&
1& 1\cr
&7A&14A&14B&21A&21C&28A&28B&28C&35A&42A&42C&56A&70A&84A&84C&105A&119A&119A\cr
} }
\vfill
The values for the first 50 head characters can be extracted from the
tables in [MS], by looking up the values of the head characters in the
monster conjugacy classes listed in the last line. The paper [MS]
also gives the decompositions of the first 50 head characters of the
monster into irreducibles, which can be compared with the next table.
(The top left corners of both tables are very similar.)
The next table gives the decomposition of the first few head
characters $H_i$ of $\hat H^0(g,V[1/2])$ into irreducible characters.
The columns correspond to the irreducible characters arranged in order
of their degrees (which are given in the first row). For example the
5th row means that the composition factors of the head representation
$H_2$ are the representations of dimension 1, 50, and 153.
\vbox{
\halign{$#$\hfil&&\ \hfil$#$\cr
&1&50&153&426&798&1029&1029&1072&1700&3654&4249&6154&6272&7497&13720&14553&17493&23324\cr
H_{-1}& 1& 0& 0& 0& 0&0&0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr H_0& 0&
0& 0& 0& 0&0&0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr H_1& 1& 1& 0& 0&
0&0&0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr H_2& 1& 1& 1& 0& 0&0&0& 0&
0& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr H_3& 2& 2& 1& 1& 0&0&0& 0& 0& 0& 0& 0&
0& 0& 0& 0& 0& 0\cr H_4& 2& 3& 2& 1& 0&0&0& 1& 0& 0& 0& 0& 0& 0& 0& 0&
0& 0\cr H_5& 4& 5& 3& 2& 1&0&0& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\cr
H_6& 4& 7& 5& 3& 1&0&0& 3& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0\cr H_7& 7& 12&
8& 7& 3&0&0& 5& 2& 1& 2& 0& 0& 0& 0& 0& 0& 0&\cr H_8& 8& 16& 13& 9&
4&0&0& 9& 4& 1& 4& 1& 0& 1& 0& 0& 0& 0&\cr H_9& 12& 25& 18& 17& 8&0&0&
15& 7& 4& 7& 1& 0& 2& 0& 1& 0& 0\cr H_{10}& 14& 35& 29& 26& 11&0&0&
27& 12& 6& 15& 4& 1& 4& 0& 2& 0& 0\cr H_{11}& 23& 53& 43& 45& 21&0&0&
43& 22& 13& 25& 7& 2& 8& 1& 4& 1& 0\cr H_{12}& 26& 75& 67& 68& 29&0&0&
74& 37& 21& 48& 16& 4& 14& 1& 9& 2& 1\cr H_{13}& 40& 114& 99& 114&
50&0&0& 119& 62& 41& 79& 27& 8& 26& 3& 18& 5& 3\cr H_{14}& 49& 161&
155& 174& 72&0&0& 202& 106& 68& 144& 58& 16& 44& 6& 32& 10& 6\cr
H_{15}& 71& 243& 233& 290&119&0&0& 328& 176& 124& 244& 99& 28& 74& 13&
58& 20& 14\cr H_{16}& 88& 348& 358& 446&173&0&0& 543& 292& 204& 422&
186& 53&124& 22&103& 35& 28\cr H_{17}& 128& 519& 543& 723&279&1&1&
876& 484& 355& 700& 319& 91&206& 43&176& 64& 54\cr H_{18}& 161& 752&
831&1121&408&1&1&1425& 785& 578&1179& 562&160&335& 72&300&110&100\cr
H_{19}& 231&1125&1263&1793&643&4&4&2280&1271& 969&1927&
937&270&542&129&500&190&180\cr H_{20}&
298&1637&1932&2769&951&6&6&3656&2043&1560&3159&1591&457&868&215&824&315&313\cr
} }
Notice that the multiplicities of nontrivial representations of small
dimension have a tendency to start off with some of the values of the
series $1, 1, 2, 3, 5, 7,\ldots$ which are the values $p(n)$ of the
partition function. The same is true for the multiplicities of
representations of the monster in the monster vertex algebra, and in
that case it can be explained using the Virasoro algebra, and the fact
that the Verma modules for the Virasoro algebra with $c=24$, $h>0$ are
irreducible and have graded dimensions $1, 1, 2, 3, 5, 7, \ldots,
p(n),\ldots $. (The irreducible factor of the Verma module with
$c=24$, $h=0$ has pieces of dimension $p(n)-p(n-1) =
1,0,1,1,2,2,4,4,7,\ldots$, which more or less accounts for the initial
multiplicities of the trivial character.) For our modular vertex
algebras this explanation does not work so well, because the Verma
modules are usually reducible over finite fields. We can get a small
amount of information by examining how the Verma modules decompose,
but this does not seem to be enough to account for why the numbers in
the table above are so similar to the numbers we get when we look at
the monster vertex algebra. On the other hand, if $g$ has
large prime order then the corresponding numbers
do not seem to be similar to those for the monster; for example,
if $g$ has order 71 then the dimensions of the $H_i$'s
start off $1,0,1,1,1,\ldots$, which cannot be the dimensions
corresponding to any representation
of the Virasoro algebra in characteristic 0.
Silly question: why do the representations of dimension 1029 appear so
late in the head characters?
\proclaim
7.~Open problems and conjectures.
\item {1.} Can the information about modular representations be used to
calculate the $|g|$-modular character tables of $C_M(g)$? The mod 7
character table of $He$ has already been worked out in [R88], so the
next simplest case is the mod 5 character table of the Harada-Norton
group $HN$. For example, by cutting up the mod 5 vertex algebra of
$HN$ using the mod 5 Virasoro algebra we find that $HN$ has
representations over ${\bf F_5}$ of dimensions 1, 133, 626 and 2451
(which are probably irreducible). Unfortunately it seems to be
difficult to get useful information like this from the later head
representations, because we run into the problem that Verma modules
over the Virasoro algebra mod $p$ are not irreducible.
\item {2.}Does the monster vertex algebra have an integral form $V$ such that
each homogeneous piece is self dual under the natural bilinear form?
It is easy to construct some monster invariant integral form by taking
some integral form and taking the intersections of its conjugates
under the action of the monster, but this will be far too small. The
following lemma shows that we are quite close to constructing such an
integral form.
\proclaim Lemma 7.1. Suppose that the spaces $V[1/2]\otimes\Z[1/6]$
and $V[1/3]\otimes\Z[1/6]$ are isomorphic as vertex algebras acted on
by the monster. Then there exists an integral form of the monster
vertex algebra with a compatible self dual integral bilinear form.
Dong and Mason [DM] have announced that these two vertex algebras are
isomorphic over the complex numbers, but their description of the
proof (which has not appeared yet) sounds as if it might be
hard to carry out over $\Z[1/6]$.
Proof. We denote $V[1/2]\otimes\Z[1/6]$ by $V[1/6]$, so we can assume
that $V[1/6]$ contains $V[1/2]$ and $V[1/3]$ as subalgebras, and we
define $V$ to be $V[1/2]\cap V[1/3]$. It is obvious that $V$ is a $\Z$-form
of the monster vertex algebra, and we just have to check that the
bilinear form on $V$ is self dual. The embeddings of $V[1/2]$ and
$V[1/3]$ into $V[1/6]$ preserve the conformal vector (as this is the
only degree 2 vector $\omega$ fixed by the monster such that the
operator $\omega_1$ multiplies every vector by its degree), so the
embeddings preserve the action of the Virasoro algebra. The bilinear
forms are determined by the grading and vertex algebra structure and
the action of the Virasoro algebra, so the embeddings also preserve
the bilinear forms on all 4 algebras. If we look at the embedding of
$V$ into $V[1/2]$ we see that the bilinear form on $V$ is self dual
over $\Z_p$ for any odd prime $p$ (as the bilinear form on $V[1/2]$ is
self dual over $\Z[1/2]$), and similarly if we look at the embedding
into $V[1/3]$ we see that the bilinear form on $V$ is self dual over
$\Z_p$ for any $p\ne 3$. Hence the symmetric bilinear form on $V$ is
self dual over all rings of $p$-adic integers, and is therefore self
dual over $\Z$. This proves lemma 7.1.
It may be possible to prove that the vertex algebras $V[1/2]$ and
$V[1/3]$ are isomorphic over $\Z[1/6]$ either by
carrying out the proof suggested in [DM] over
$\Z[1/6]$, or by constructing the monster
vertex algebra as a sum of eigenspaces of an element of the monster of
type 6B. This element corresponds to a fixed point free element of
order 6 in $\Aut(\Lambda)$ (of trace 12) whose cube and square are the
elements of orders $2$ and $3$ in $\Aut(\Lambda)$ used to construct
$V[1/2]$ and $V[1/3]$.
The integral form $V$ would give integral forms for all
the homogeneous spaces $V_n$, and in particular would give an
integral form on the Griess algebra $V_1$. This cannot be the same as the
integral form constructed by Conway and Norton
in [C85], because the one in [C85] contains
the element $\omega/2$ (which is denoted by 1 there). It seems possible
that Conway and Norton's integral form is spanned by $\omega/2$ together with
the elements of $V_1$ which have integral inner product with $\omega/2$.
Notice that the bilinear form used in [C85] is half the bilinear form
on $V_1$.
\item{3.}Assume the integral form $V$ exists.
Is $\hat H^1( g,V)$ zero whenever $g$ is an element of the monster
whose Hauptmodul has no negative coefficients? (Theorem 4.7 shows this
for some elements of odd order.) If the coefficients of the Hauptmodul
for $g$ alternate in sign, do the groups $\hat H^0(g,V_{2n}) $ and
$\hat H^1(g,V_{2n+1})$ vanish? (Theorem 5.3 proves this for elements
of type 2B.)
\item{4.} What happens if $g$ is an element of composite order?
For example, if we look at the Hauptmodul for an element of type 4B we
see that it starts off $q^{-1}+52q+834q^3+4760q^5$, and the
coefficient $52$ of $q^1$ is the dimension of the Lie algebra $F_4$,
and the centralizer of an element of type 4B is of the form $(4\times
F_4({\bf F_2})).2$. This suggests there should be a modular vertex
algebra corresponding to elements of type 4B, whose ``compression''
should be the reduction mod 2 or 4 of a vertex algebra for $F_4$
defined over $\Z$, in the same way that the compression of the vertex
algebra for elements of type 3C is the reduction mod 3 of an algebra
for $E_8$. (See the end of section 4.)
\item {5.} We can construct Lie algebras and superalgebras
which have much the same relation to our modular vertex algebras as
the monster Lie algebra [B92] has to the monster vertex algebra. We
do this by using the $\Z_p$-forms of the monster vertex algebra to put
$\Z_p$-forms on the monster Lie algebra (with a self dual symmetric
invariant bilinear form), and then take the Tate cohomology of this
$\Z_p$-form of the monster Lie algebra, which by the comments at the
beginning of section 2 produces a Lie superalgebra. (Of course if we
have a good integral form of the monster Lie algebra we can use this
directly and not worry about $\Z_p$-forms.) The Lie algebras and
superalgebras we get are similar to generalized Kac-Moody
algebras, except that they are over fields of characteristic $p$
rather than characteristic 0: they have a root system, a Cartan
subalgebra, an invariant nonsingular symmetric bilinear form, a Cartan
involution, and a $\Z$-grading with finite dimensional homogeneous
pieces. Their structure as $C_M(g)$ modules can be described as
follows: they have a $\Z^2$-grading, such that the piece of degree
$(m,n)$ is isomorphic to $^gV_{mn}$ if $(m,n)\ne (0,0)$, and the piece
of degree $(0,0)$ (the Cartan subalgebra) is 2-dimensional and acted
on trivially by $C_M(g)$. This suggests that there should be some sort
of theory of ``generalized Kac-Moody algebras $\bmod p$'', which could
be applied to study these algebras. For example, we could ask for the
Lie algebra homology groups of the positive degree subalgebras (which
in characteristic 0 is equivalent to asking for the simple roots).
Notice that many of these modular Lie algebras cannot be obtained by
reducing some integral form of a generalized Kac-Moody algebra $\bmod
p$, because the denominator formula shows that some of the simple
roots would then have negative multiplicity.
\item {6.} The modules $\hat H^*(g,V[1/2])$ are acted on not just
by the centralizer of $g$ but by the normalizer $N_M(g)$ of $g$. What
are the traces of elements of the normalizer of $g$ that are not in
the centralizer? It may be possible to do this by extending
proposition 2.2 to the case when the subgroup $\langle g \rangle$
is only normal and not central.
(Notice that if $\hat H^1(g,V[1/2])$ is nonzero then
elements of the normalizer do not necessarily preserve the algebra
structure on $\hat H^*(g,V[1/2])$.)
\item {7.} Give a complete proof of assertion 5.1.
\item{8.} Can the modular vertex algebras $^gV$ be lifted to characteristic $0$
in some way? The answer to the strong form of this question is usually
``no'': it is easy to check that it is usually impossible to lift
$^gV$ to a vertex algebra in characteristic 0 that is acted on by
$C_M(g)$. However Queen [Q] found strong evidence that $^gV$ could be
lifted to some $C_M(g)$ representation in characteristic 0 (which
cannot carry an invariant vertex algebra structure). The
representations that Queen found evidence for are now called twisted
sectors. Perhaps these representations have some sort of ``twisted''
vertex algebra structure, where the vertex operators $a(z)$ have
branch points of order $p$ at the origin. It might be possible to use
some sort of analogue of Witt vectors for vertex algebras to construct
these. (But there is one serious obstruction to any canonical way of
lifting some $^gV$'s to characteristic 0: the automorphism group in
characteristic 0 is sometimes a {\it nonsplit} central extension of
the automorphism group in characteristic $p$.) On the other hand we
have seen (in the remarks at the end of sections 4 and 5) that if $g$
is of type 3C or 2B then $^gV$ can probably be lifted to a vertex
algebra in characteristic 0, which is not acted on by $C_M(g)$.
Dong, Li and Mason [DLM] have recently made some progress on this question
by constructing a twisted sector that is probably a lift to characteristic
zero of the space $^gV$ when $g$ is of type $2A$.
\item{9.} If $p=3,5,7$, or $13$ and $g$ is an element of type
$pB$ in the monster corresponding to the group $\Gamma_0(p)$ then the
group $C_M(g)/O_p(C_M(g))$ contains an element of order 2 in its center. We
conjecture that this element of order 2 acts as $+1$ on $\hat
H^0(g,V[1/2])$ and as $-1$ on $\hat H^1(g,V[1/2])$. This would imply
that the graded modular characters of both $\hat H^0$ and $\hat H^1$
can be expressed as a linear combination of 2 Hauptmoduls. These are
the only elements of prime order in the monster not already covered by
theorems 4.7, 5.2, 5.3, and question 3 above, so an affirmative answer
to this question and question 3 would mean that we would have a
complete description of the modular characters of both $\hat H^0(g,V)$
and $\hat H^1(g,V)$ for all elements $g$ of prime order in the
monster.
\proclaim References.
\item{[AW]} M. F. Atiyah, C. T. C. Wall, Cohomology of finite groups,
in ``Algebraic number theory'', editors J.~W.~S.~Cassels and
A.~Fr\"ohlich, Academic Press 1967.
\item{[B86]}{R. E. Borcherds, Vertex algebras, Kac-Moody algebras,
and the monster. Proc. Natl. Acad. Sci. USA. Vol. 83 (1986)
3068-3071.}
\item{[B92]}{R. E. Borcherds,
Monstrous moonshine and monstrous Lie superalgebras, Invent. Math.
109, 405-444 (1992).}
\item{[C]}{J. H. Conway, R. T. Curtis, S. P. Norton,
R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon
Press, Oxford, 1985.}
\item{[C85]} J. H. Conway, A simple construction for the Fischer-Griess
monster group, Invent. Math. 79 (1985) p. 513-540.
\item{[CN]}{J. H. Conway, S. Norton, Monstrous moonshine,
Bull. London. Math. Soc. 11 (1979) 308-339.}
\item{[CR]} C.~W.~Curtis and I.~Reiner,
``Methods of representation theory Vol 1'', Wiley Interscience, 1981
and 1990.
\item{[DLM]} C. Dong, H. Li, G. Mason, Some twisted sectors for the moonshine
module, to appear in Contemporary Math.
\item{[DM]}C.~Dong and G.~Mason,
On the construction of the moonshine module as a
$\Z_p$-orbifold,
Santa Cruz preprint, 1992.
To appear in: Proc. 1992 Joint
Summer Research Conference on Conformal Field Theory, Topological Field theory
and Quantum Groups, Mount Holyoke, 1992, Contemporary Math.
\item{[FLM]}{I. B. Frenkel, J. Lepowsky, A. Meurman,
Vertex operator algebras and the monster, Academic press 1988. }
\item{[Mac]} I. G. Macdonald, ``Symmetric functions and Hall polynomials'',
Oxford University press, 1979.
\item{[MS]} J. McKay, H. Strauss, The $q$-series of monstrous moonshine
and the decomposition of the head characters, Comm. in Alg. (1990) 18,
253-278.
\item{[M]} P. Montague, Third and Higher Order NFPA Twisted
Constructions of Conformal Field Theories from Lattices, preprint,
submitted to Nuc. Phys. B.
\item{[N]}{S. P. Norton, Generalized Moonshine,
Proc. Symp. Pure Math. 47 (1987) p. 208-209.}
\item{[Q]} L. Queen, Some relations between finite groups, Lie groups,
and modular functions, PhD thesis, Cambridge University, England, 1980.
\item{[R88]}{A. J. E. Ryba, Calculation of the 7-modular characters
of the Held group. J. Algebra, 117, 240-255, 1988. }
\item{[R94]}{A. J. E. Ryba, Modular Moonshine?, 1994 preprint. }
\end