Pariah moonshine and math-writing

Getting mathematics into Nature (the journal) is next to impossible. Ask David Mumford and John Tate about it.

Last month, John Duncan, Michael Mertens and Ken Ono managed to do just that.

Inevitably, they had to suffer through a photoshoot and give their university’s PR-people some soundbites.

CAPTION

In the simplest terms, an elliptic curve is a doughnut shape with carefully placed points, explain Emory University mathematicians Ken Ono, left, and John Duncan, right. “The whole game in the math of elliptic curves is determining whether the doughnut has sprinkles and, if so, where exactly the sprinkles are placed,” Duncan says.

CAPTION

“Imagine you are holding a doughnut in the dark,” Emory University mathematician Ken Ono says. “You wouldn’t even be able to decide whether it has any sprinkles. But the information in our O’Nan moonshine allows us to ‘see’ our mathematical doughnuts clearly by giving us a wealth of information about the points on elliptic curves.”

(Photos by Stephen Nowland, Emory University. See here and here.)

Some may find this kind of sad, or a bad example of over-popularisation.

I think they do a pretty good job of getting the notion of rational points on elliptic curves across.

That’s what the arithmetic of elliptic curves is all about, finding structure in patterns of sprinkles on special doughnuts. And hey, you can get rich and famous if you’re good at it.

Their Nature-paper Pariah moonshine is a must-read for anyone aspiring to write a math-book aiming at a larger audience.

It is an introduction to and a summary of the results they arXived last February O’Nan moonshine and arithmetic.

Update (October 21st)

John Duncan send me this comment via email:

“Strictly speaking the article was published in Nature Communications (https://www.nature.com/ncomms/). We were also rejected by Nature. But Nature forwarded our submission to Nature Communications, and we had a great experience. Specifically, the review period was very fast (compared to most math journals), and the editors offered very good advice.

My understanding is that Nature Communications is interested in publishing more pure mathematics. If someone reading this has a great mathematical story to tell, I (humbly) recommend to them this option. Perhaps the work of Mumford–Tate would be more agreeably received here.

By the way, our Nature Communications article is open access, available at https://www.nature.com/articles/s41467-017-00660-y.”

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.