%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