Moonshine for everyone

Today, Samuel Dehority, Xavier Gonzalez, Neekon Vafa and Roger Van Peski arXived their paper Moonshine for all finite groups.

Originally, Moonshine was thought to be connected to the Monster group. McKay and Thompson observed that the first coefficients of the normalized elliptic modular invariant

J(\tau) = q^{-1} + 196884 q + 21493760 q^2 + 864229970 q^3 + \ldots

could be written as sums of dimensions of the first few irreducible representations of the monster group:

1=1,~\quad 196884=196883+1,~\quad 21493760=1+196883+21296876,~\quad … \]

Soon it transpired that there ought to be an infinite dimensional graded vectorspace, the moonshine module

V^{\sharp} = \bigoplus_{n=-1}^{\infty}~V^{\sharp}_n \]

with every component $V^{\sharp}_n$ being a representation of the monster group $\mathbb{M}$ of which the dimension coincides with the coefficient of $q^n$ in $J(\tau)$.

It only got better, for any conjugacy class $[ g ]$ of the monster, if you took the character series

T_g(\tau) = \sum_{n=-1}^{\infty} Tr(g | V^{\sharp}_n) q^n \]

you get a function invariant under the action of the subgroup

\Gamma_0(n) = \{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}~:~c = 0~mod~n \} \]

acting via transformations $\tau \mapsto \frac{a \tau + b}{c \tau + d}$ on the upper half plane where $n$ is the order of $g$ (or, for the experts, almost).

Soon, further instances of ‘moonshine’ were discovered for other simple groups, the unifying feature being that one associates to a group $G$ a graded representation $V$ such that the character series of this representation for an element $g \in G$ is an invariant modular function with respect to the subgroup $\Gamma_0(n)$ of the modular group, with $n$ being the order of $g$.

Today, this group of people proved that there is ‘moonshine’ for any finite group whatsoever.

They changed the definition of moonshine slightly to introduce the notion of moonshine of depth $d$ which meant that they want the dimension sequence of their graded module to be equal to $J(\tau)$ under the action of the normalized $d$-th Hecke operator, which means equal to

\sum_{ac=d,0 \leq b < c} J(\frac{a \tau + b}{c}) \]
as they are interested in the asymptotic behaviour of the components $V_n$ with respect to the regular representation of $G$.

What baffled me was their much weaker observation (remark 2) saying that you get ‘moonshine’ in the form described above, that is, a graded representation $V$ such that for every $g \in G$ you get a character series which is invariant under $\Gamma_0(n)$ with $n=ord(g)$ (and no smaller divisor of $n$), for every finite group $G$.

And, more importantly, you can explain this to any student taking a first course in group theory as all you need is Cayley’s theorem stating that any finite group is a subgroup of some symmetric group $S_n$.

Here’s the idea: take the original monster-moonshine module $V^{\sharp}$ but forget all about the action of $\mathbb{M}$ (that is, consider it as a plain vectorspace) and consider the graded representation

V = (V^{\sharp})^{\otimes n} \]

with the natural action of $S_n$ on the tensor product.

Now, embed a la Cayley $G$ into $S_n$ then you know that the order of $g \in G$ is the least common multiple of the cycle lengths of the permutation it it send to. Now, it is fairly trivial to see that the character series of $V$ with respect to $g$ (having cycle lengths $(k_1,k_2,\dots,k_l)$, including cycles of length one) is equal to the product

J(k_1 \tau) J(k_2 \tau) \dots J(k_l \tau) \]

which is invariant under $\Gamma_0(n)$ with $n = lcm(k_i)$ (but no $\Gamma_0(m)$ with $m$ a proper divisor of $n$).

For example, for $G=S_4$ we have as character series of $(V^{\sharp})^{\otimes 4}$

(1)(2)(3)(4) \mapsto J(\tau)^4 \]

(12)(3)(4) \mapsto J(2 \tau) J(\tau)^2 \]

(12)(34) \mapsto J(2 \tau)^2 \]

(123)(4) \mapsto J(3 \tau) J(\tau) \]

(1234) \mapsto J(4 \tau) \]

Clearly, the main results of the paper are much more subtle, but I’m already happy with this version of ‘moonshine for everyone’!

Monsters and Moonshine : a booklet

I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.

The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu group $M_{12}$. This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’. We discuss Galois’ last letter and the simple groups $L_2(5),L_2(7)$ and $L_2(11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered ‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover what the largest sporadic simple group, the Monster-group, sees of the modular group.

The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $PSL_2(\mathbb{Z})$. We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using Farey codes and use those to give a series of simple groups including as special members $L_2(5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E(8)$ in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s solution of the $E(8)$-observation.

I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!

If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to the “field with one element” and its connections to (noncommutative) geometry and number theory.

I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so far only exists as prep-notes. The links above will always link to the latest versions of these booklets.

E(8) from moonshine groups

Are the valencies of the 171 moonshine groups are compatible, that is, can one construct a (disconnected) graph on the 171 vertices such that in every vertex (determined by a moonshine group G) the vertex-valency coincides with the valency of the corresponding group? Duncan describes a subset of 9 moonshine groups for which the valencies are compatible. These 9 groups are characterized as those moonshine groups G
having width 1 at the cusp and such that their intersection with the modular group is big.

Time to wrap up this series on John Duncan‘s paper Arithmetic groups and the affine E8 Dynkin diagram in which he gives a realization of the extended E(8)-Dynkin diagram (together with its isotropic root vector) from the moonshine groups, compatible with McKay’s E(8)-observation.

In the previous post we have described all 171 moonshine groups using Conway’s big picture. This description will allow us to associate two numbers to a moonshine group $G \subset PSL_2(\mathbb{R}) $.
Recall that for any such group we have a positive integer $N $ such that

$\Gamma_0(N) \subset G \subset \Gamma_0(h,\frac{N}{h})+ $

where $h $ is the largest divisor of 24 such that $h^2 | N $. Let us call $n_G=\frac{N}{h} $ the dimension of $G $ (Duncan calls this number the ‘normalized level’) as it will give us the dimension component at the vertex determined by $G $.

We have also seen last time that any moonshine group is of the form $G = \Gamma_0(n_G || h)+e,f,g $, that is, $G/\Gamma_0(n_G ||h) $ is an elementary abelian group $~(\mathbb{Z}/2\mathbb{Z})^m $ generated by Atkin-Lehner involutions. Let’s call $v_G=m+1 $ the valency of the group $G $ as it will give s the valency of the vertex determined by $G $.

It would be nice to know whether the valencies of the 171 moonshine groups are compatible, that is, whether one can construct a (disconnected) graph on the 171 vertices such that in each vertex (determined by a moonshine group $G $) the vertex-valency coincides with the valency of the corresponding group.

Duncan describes a subset of 9 moonshine groups for which the valencies are compatible. These 9 groups are characterized as those moonshine groups $G $
having width 1 at the cusp and such that their intersection with the modular group $\Gamma = PSL_2(\mathbb{Z}) $ is big, more precisely the index $[\Gamma : \Gamma \cap G] \leq 12 $ and $[\Gamma : \Gamma \cap G]/[G : \Gamma \cap G] \leq 3 $.

They can be described using the mini-moonshine picture on the right. They are :

The modular group itself $1=\Gamma $, being the stabilizer of the lattice 1. This group has clearly dimension and valency equal to one.

The modular subgroup $2=\Gamma_0(2) $ being the point-wise stabilizer of the lattices 1 and 2 (so it has valency one and dimension two, and, its normalizer $2+ =\Gamma_0(2)+ $ which is the set-wise stabilizer of the lattices 1 and 2 and the one Atkin-Lehner involution interchanges both. So, this group has valency two (as we added one involution) as well as dimension two.

Likewise, the groups $3+=\Gamma_0(3)+ $ and $5+=\Gamma_0(5)+ $ are the stabilzer subgroups of the red 1-cell (1,3) resp. the green 1-cell (1,5) and hence have valency two (as we add one involution) and dimensions 3 resp. 5.

The group $4+=\Gamma_0(4)+ $ stabilizes the (1|4)-thread and as we add one involution must have valency 2 and dimension 4.

On the other hand, the group $6+=\Gamma_0(6)+ $ stabilizes the unique 2-cell in the picture (having lattices 1,2,3,6) so this time we will add three involutions (horizontal and vertical switches and their product the antipodal involution). Hence, for this group the valency is three and its dimension is equal to six.

Remain the two groups connected to the mini-snakes in the picture. The red mini-snake (top left hand) is the ball with center 3 and hyperdistance 3 and determines the group $3||3=\Gamma_0(3||3) $ which has valency one (we add no involutions) and dimension 3. The blue mini-snake (the extended D(5)-Dynkin in the lower right corner) determines the group $4||2+=\Gamma(4||2)+ $ which has valency two and dimension 4.

The valencies of these 9 moonshine groups are compatible and they can be arranged in the extended E(8) diagram depicted below

Moreover, the dimensions of the groups give the exact dimension-components of the isotropic root of the extended E(8)-diagram. Further, the dimension of the group is equal to the order of the elements making up the conjugacy class of the monster to which exactly the given groups correspond via monstrous moonshine and hence compatible with John McKay‘s original E(8)-observation!

Once again, I would love to hear when someone has more information on the cell-decomposition of the moonshine picture or if someone can extend the moonshine E(8)-graph, possibly to include all 171 moonshine groups.

looking for the moonshine picture

We have seen that Conway’s big picture helps us to determine all arithmetic subgroups of $PSL_2(\mathbb{R}) $ commensurable with the modular group $PSL_2(\mathbb{Z}) $, including all groups of monstrous moonshine.

As there are exactly 171 such moonshine groups, they are determined by a finite subgraph of Conway’s picture and we call the minimal such subgraph the moonshine picture. Clearly, we would like to determine its structure.

On the left a depiction of a very small part of it. It is the minimal subgraph of Conway’s picture needed to describe the 9 moonshine groups appearing in Duncan’s realization of McKay’s E(8)-observation. Here, only three primes are relevant : 2 (blue lines), 3 (reds) and 5 (green). All lattices are number-like (recall that $M \frac{g}{h} $ stands for the lattice $\langle M e_1 + \frac{g}{h} e_2, e_2 \rangle $).

We observe that a large part of this mini-moonshine picture consists of the three p-tree subgraphs (the blue, red and green tree starting at the 1-lattice $1 = \langle e_1,e_2 \rangle $. Whereas Conway’s big picture is the product over all p-trees with p running over all prime numbers, we observe that the mini-moonshine picture is a very small subgraph of the product of these three subtrees. In fact, there is just one 2-cell (the square 1,2,6,3).

Hence, it seems like a good idea to start our investigation of the full moonshine picture with the determination of the p-subtrees contained in it, and subsequently, worry about higher dimensional cells constructed from them. Surely it will be no major surprise that the prime numbers p that appear in the moonshine picture are exactly the prime divisors of the order of the monster group, that is p=2,3,5,7,11,13,17,19,23,29,31,41,47,59 or 71. Before we can try to determine these 15 p-trees, we need to know more about the 171 moonshine groups.

Recall that the proper way to view the modular subgroup $\Gamma_0(N) $ is as the subgroup fixing the two lattices $L_1 $ and $L_N $, whence we will write $\Gamma_0(N)=\Gamma_0(N|1) $, and, by extension we will denote with $\Gamma_0(X|Y) $ the subgroup fixing the two lattices $L_X $ and $L_Y $.

As $\Gamma_0(N) $ fixes $L_1 $ and $L_N $ it also fixes all lattices in the (N|1)-thread, that is all lattices occurring in a shortest path from $L_1 $ to $L_N $ (on the left a picture of the (200|1)-thread).

If $N=p_1^{a_1} p_2^{a_2} \ldots p_k^{a_k} $, then the (N|1)-thread has $2^k $ involutions as symmetries, called the Atkin-Lehner involutions. For every exact divisor $e || N $ (that is, $e|N $ and $gcd(e,\frac{N}{e})=1 $ we have an involution $W_e $ which acts by sending each point in the thread-cell corresponding to the prime divisors of $e $ to its antipodal cell-point and acts as the identity on the other prime-axes. For example, in the (200|1)-thread on the left, $W_8 $ is the left-right reflexion, $W_{25} $ the top-bottom reflexion and $W_{200} $ the antipodal reflexion. The set of all exact divisors of N becomes the group $~(\mathbb{Z}/2\mathbb{Z})^k $ under the operation $e \ast f = \frac{e \times f}{gcd(e,f)^2} $.

Most of the moonshine groups are of the form $\Gamma_0(n|h)+e,f,g,… $ for some $N=h.n $ such that $h | 24 $ and $h^2 | N $. The group $\Gamma_0(n|h) $ is then conjugate to the modular subgroup $\Gamma_0(\frac{n}{h}) $ by the element $\begin{bmatrix} h & 0 \ 0 & 1 \end{bmatrix} $. With $\Gamma_0(n|h)+e,f,g,… $ we mean that the group $\Gamma_0(n|h) $ is extended with the involutions $W_e,W_f,W_g,… $. If we simply add all Atkin-Lehner involutions we write $\Gamma_0(n|h)+ $ for the resulting group.

Finally, whenever $h \not= 1 $ there is a subgroup $\Gamma_0(n||h)+e,f,g,… $ which is the kernel of a character $\lambda $ being trivial on $\Gamma_0(N) $ and on all involutions $W_e $ for which every prime dividing $e $ also divides $\frac{n}{h} $, evaluating to $e^{\frac{2\pi i}{h}} $ on all cosets containing $\begin{bmatrix} 1 & \frac{1}{h} \ 0 & 1 \end{bmatrix} $ and to $e^{\pm \frac{2 \pi i }{h}} $ for cosets containing $\begin{bmatrix} 1 & 0 \ n & 0 \end{bmatrix} $ (with a + sign if $\begin{bmatrix} 0 & -1 \ N & 0 \end{bmatrix} $ is present and a – sign otherwise). Btw. it is not evident at all that this is a character, but hard work shows it is!

Clearly there are heavy restrictions on the numbers that actually occur in moonshine. In the paper On the discrete groups of moonshine, John Conway, John McKay and Abdellah Sebbar characterized the 171 arithmetic subgroups of $PSL_2(\mathbb{R}) $ occuring in monstrous moonshine as those of the form $G = \Gamma_0(n || h)+e,f,g,… $ which are

  • (a) of genus zero, meaning that the quotient of the upper-half plane by the action of $G \subset PSL_2(\mathbb{R}) $ by Moebius-transformations gives a Riemann surface of genus zero,
  • (b) the quotient group $G/\Gamma_0(nh) $ is a group of exponent 2 (generated by some Atkin-Lehner involutions), and
  • (c) every cusp can be mapped to $\infty $ by an element of $PSL_2(\mathbb{R}) $ which conjugates the group to one containing $\Gamma_0(nh) $.

Now, if $\Gamma_0(n || h)+e,f,g,… $ is of genus zero, so is the larger group $\Gamma_0(n | h)+e,f,g,… $, which in turn, is conjugated to the group $\Gamma_0(\frac{n}{h})+e,f,g,… $. Therefore, we need a list of all groups of the form $\Gamma_0(\frac{n}{h})+e,f,g,… $ which are of genus zero. There are exactly 123 of them, listed on the right.

How does this help to determine the structure of the p-subtree of the moonshine picture for the fifteen monster-primes p? Look for the largest p-power $p^k $ such that $p^k+e,f,g… $ appears in the list. That is for p=2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 these powers are resp. 5,3,2,2,1,1,1,1,1,1,1,1,1,1,1. Next, look for the largest p-power $p^l $ dividing 24 (that is, 3 for p=2, 1 for p=3 and 0 for all other primes). Then, these relevant moonshine groups contain the modular subgroup $\Gamma_0(p^{k+2l}) $ and are contained in its normalizer in $PSL_2(\mathbb{R}) $ which by the Atkin-Lehner theorem is precisely the group $\Gamma_0(p^{k+l}|p^l)+ $.

Right, now the lattices fixed by $\Gamma_0(p^{k+2l}) $ (and permuted by its normalizer), that is the lattices in our p-subtree, are those that form the $~(p^{k+2l}|1) $-snake in Conway-speak. That is, the lattices whose hyper-distance to the $~(p^{k+l}|p^l) $-thread divides 24. So for all primes larger than 2 or 3, the p-tree is just the $~(p^l|1) $-thread.

For p=3 the 3-tree is the (243|1)-snake having the (81|3)-thread as its spine. It contains the following lattices, all of which are number-like.

Depicting the 2-tree, which is the (2048|1)-snake may take a bit longer… Perhaps someone should spend some time figuring out which cells of the product of these fifteen trees make up the moonshine picture!