Tag: monster

For the better part of the 30ties, Ernst Witt (1) did hang out with the rest of the ‘Noetherknaben’, the group of young mathematicians around Emmy Noether (3) in Gottingen.

In 1934 Witt became Helmut Hasse‘s assistent in Gottingen, where he qualified as a university lecturer in 1936. By 1938 he has made enough of a name for himself to be offered a lecturer position in Hamburg and soon became an associate professor, the down-graded position held by Emil Artin (2) until he was forced to emigrate in 1937.

A former fellow student of him in Gottingen, Erna Bannow (4), had gone earlier to Hamburg to work with Artin. She continued her studies with Witt and finished her Ph.D. in 1939. In 1940 Erna Bannow and Witt married.

So, life was smiling on Ernst Witt that sunday january 28th 1940, both professionally and personally. There was just one cloud on the horizon, and a rather menacing one. He was called up by the Wehrmacht and knew he had to enter service in february. For all he knew, he was spending the last week-end with his future wife… (later in february 1940, Blaschke helped him to defer his military service by one year).

Still, he desperately wanted to finish his paper before entering the army, so he spend most of that week-end going through the final version and submitted it on monday, as the published paper shows.

In the 70ties, Witt suddenly claimed he did discover the Leech lattice ${\Lambda}$ that sunday. Last time we have seen that the only written evidence for Witt’s claim is one sentence in his 1941-paper Eine Identität zwischen Modulformen zweiten Grades. “Bei dem Versuch, eine Form aus einer solchen Klassen wirklich anzugeben, fand ich mehr als 10 verschiedene Klassen in ${\Gamma_{24}}$.”

But then, why didn’t Witt include more details of this sensational lattice in his paper?

Ina Kersten recalls on page 328 of Witt’s collected papers : “In his colloquium talk “Gitter und Mathieu-Gruppen” in Hamburg on January 27, 1970, Witt said that in 1938, he had found nine lattices in ${\Gamma_{24}}$ and that later on January 28, 1940, while studying the Steiner system ${S(5,8,24)}$, he had found two additional lattices ${M}$ and ${\Lambda}$ in ${\Gamma_{24}}$. He continued saying that he had then given up the tedious investigation of ${\Gamma_{24}}$ because of the surprisingly low contribution

$\displaystyle | Aut(\Lambda) |^{-1} < 10^{-18}$

to the Minkowski density and that he had consented himself with a short note on page 324 in his 1941 paper.”

In the last sentence he refers to the fact that the sum of the inverse orders of the automorphism groups of all even unimodular lattices of a given dimension is a fixed rational number, the Minkowski-Siegel mass constant. In dimension 24 this constant is

$\displaystyle \sum_{L} \frac{1}{| Aut(L) |} = \frac {1027637932586061520960267}{129477933340026851560636148613120000000} \approx 7.937 \times 10^{-15}$

That is, Witt was disappointed by the low contribution of the Leech lattice to the total constant and concluded that there might be thousands of new even 24-dimensional unimodular lattices out there, and dropped the problem.

If true, the story gets even better : not only claims Witt to have found the lattices ${A_1^{24}=M}$ and ${\Lambda}$, but also enough information on the Leech lattice in order to compute the order of its automorphism group ${Aut(\Lambda)}$, aka the Conway group ${Co_0 = .0}$ the dotto-group!

Is this possible? Well fortunately, the difficulties one encounters when trying to compute the order of the automorphism group of the Leech lattice from scratch, is one of the better documented mathematical stories around.

The books From Error-Correcting Codes through Sphere Packings to Simple Groups by Thomas Thompson, Symmetry and the monster by Mark Ronan, and Finding moonshine by Marcus du Sautoy tell the story in minute detail.

It took John Conway 12 hours on a 1968 saturday in Cambridge to compute the order of the dotto group, using the knowledge of Leech and McKay on the properties of the Leech lattice and with considerable help offered by John Thompson via telephone.

But then, John Conway is one of the fastest mathematicians the world has known. The prologue of his book On numbers and games begins with : “Just over a quarter of a century ago, for seven consecutive days I sat down and typed from 8:30 am until midnight, with just an hour for lunch, and ever since have described this book as “having been written in a week”.”

Conway may have written a book in one week, Ernst Witt did complete his entire Ph.D. in just one week! In a letter of August 1933, his sister told her parents : “He did not have a thesis topic until July 1, and the thesis was to be submitted by July 7. He did not want to have a topic assigned to him, and when he finally had the idea, he started working day and night, and eventually managed to finish in time.”

So, if someone might have beaten John Conway in fast-computing the dottos order, it may very well have been Witt. Sadly enough, there is a lot of circumstantial evidence to make Witt’s claim highly unlikely.

For starters, psychology. Would you spend your last week-end together with your wife to be before going to war performing an horrendous calculation?

Secondly, mathematical breakthroughs often arise from newly found insight. At that time, Witt was also working on his paper on root lattices “Spiegelungsgrupen and Aufzähling halbeinfacher Liescher Ringe” which he eventually submitted in january 1941. Contained in that paper is what we know as Witt’s lemma which tells us that for any integral lattice the sublattice generated by vectors of norms 1 and 2 is a direct sum of root lattices.

This leads to the trick of trying to construct unimodular lattices by starting with a direct sum of root lattices and ‘adding glue’. Although this gluing-method was introduced by Kneser as late as 1967, Witt must have been aware of it as his 16-dimensional lattice ${D_{16}^+}$ is constructed this way.

If Witt wanted to construct new 24-dimensional even unimodular lattices in 1940, it would be natural for him to start off with direct sums of root lattices and trying to add vectors to them until he got what he was after. Now, all of the Niemeier-lattices are constructed this way, except for the Leech lattice!

I’m far from an expert on the Niemeier lattices but I would say that Witt definitely knew of the existence of ${D_{24}^+}$, ${E_8^3}$ and ${A_{24}^+}$ and that it is quite likely he also constructed ${(D_{16}E_8)^+, (D_{12}^2)^+, (A_{12}^2)^+, (D_8^3)^+}$ and possibly ${(A_{17}E_7)^+}$ and ${(A_{15}D_9)^+}$. I’d rate it far more likely Witt constructed another two such lattices on sunday january 28th 1940, rather than discovering the Leech lattice.

Finally, wouldn’t it be natural for him to include a remark, in his 1941 paper on root lattices, that not every even unimodular lattices can be obtained from sums of root lattices by adding glue, the Leech lattice being the minimal counter-example?

If it is true he was playing around with the Steiner systems that sunday, it would still be a pretty good story he discovered the lattices ${(A_2^{12})^+}$ and ${(A_1^{24})^+}$, for this would mean he discovered the Golay codes in the process!

Which brings us to our next question : who discovered the Golay code?

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.

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!

Conway and Norton showed that there are exactly 171 moonshine functions and associated two arithmetic subgroups to them. We want a tool to describe these and here’s where Conway’s big picture comes in very handy. All moonshine groups are arithmetic groups, that is, they are commensurable with the modular group. Conway’s idea is to view several of these groups as point- or set-wise stabilizer subgroups of finite sets of (projective) commensurable 2-dimensional lattices.

Expanding (and partially explaining) the original moonshine observation of McKay and Thompson, John Conway and Simon Norton formulated monstrous moonshine :

To every cyclic subgroup $\langle m \rangle$ of the Monster $\mathbb{M}$ is associated a function

$f_m(\tau)=\frac{1}{q}+a_1q+a_2q^2+\ldots$ with $q=e^{2 \pi i \tau}$ and all coefficients $a_i \in \mathbb{Z}$ are characters at $m$ of a representation of $\mathbb{M}$. These representations are the homogeneous components of the so called Moonshine module.

Each $f_m$ is a principal modulus for a certain genus zero congruence group commensurable with the modular group $\Gamma = PSL_2(\mathbb{Z})$. These groups are called the moonshine groups.

Conway and Norton showed that there are exactly 171 different functions $f_m$ and associated two arithmetic subgroups $F(m) \subset E(m) \subset PSL_2(\mathbb{R})$ to them (in most cases, but not all, these two groups coincide).

Whereas there is an extensive literature on subgroups of the modular group (see for instance the series of posts starting here), most moonshine groups are not contained in the modular group. So, we need a tool to describe them and here’s where Conway’s big picture comes in very handy.

All moonshine groups are arithmetic groups, that is, they are subgroups $G$ of $PSL_2(\mathbb{R})$ which are commensurable with the modular group $\Gamma = PSL_2(\mathbb{Z})$ meaning that the intersection $G \cap \Gamma$ is of finite index in both $G$ and in $\Gamma$. Conway’s idea is to view several of these groups as point- or set-wise stabilizer subgroups of finite sets of (projective) commensurable 2-dimensional lattices.

Start with a fixed two dimensional lattice $L_1 = \mathbb{Z} e_1 + \mathbb{Z} e_2 = \langle e_1,e_2 \rangle$ and we want to name all lattices of the form $L = \langle v_1= a e_1+ b e_2, v_2 = c e_1 + d e_2 \rangle$ that are commensurable to $L_1$. Again this means that the intersection $L \cap L_1$ is of finite index in both lattices. From this it follows immediately that all coefficients $a,b,c,d$ are rational numbers.

It simplifies matters enormously if we do not look at lattices individually but rather at projective equivalence classes, that is $~L=\langle v_1, v_2 \rangle \sim L’ = \langle v’_1,v’_2 \rangle$ if there is a rational number $\lambda \in \mathbb{Q}$ such that $~\lambda v_1 = v’_1, \lambda v_2=v’_2$. Further, we are of course allowed to choose a different ‘basis’ for our lattices, that is, $~L = \langle v_1,v_2 \rangle = \langle w_1,w_2 \rangle$ whenever $~(w_1,w_2) = (v_1,v_2).\gamma$ for some $\gamma \in PSL_2(\mathbb{Z})$.
Using both operations we can get any lattice in a specific form. For example,

$\langle \frac{1}{2}e_1+3e_2,e_1-\frac{1}{3}e_2 \overset{(1)}{=} \langle 3 e_1+18e_2,6e_1-2e_2 \rangle \overset{(2)}{=} \langle 3 e_1+18 e_2,38 e_2 \rangle \overset{(3)}{=} \langle \frac{3}{38}e_1+\frac{9}{19}e_2,e_2 \rangle$

Here, identities (1) and (3) follow from projective equivalence and identity (2) from a base-change. In general, any lattice $L$ commensurable to the standard lattice $L_1$ can be rewritten uniquely as $L = \langle Me_1 + \frac{g}{h} e_2,e_2 \rangle$ where $M$ a positive rational number and with $0 \leq \frac{g}{h} < 1$.

Another major feature is that one can define a symmetric hyper-distance between (equivalence classes of) such lattices. Take $L=\langle Me_1 + \frac{g}{h} e_2,e_2 \rangle$ and $L’=\langle N e_1 + \frac{i}{j} e_2,e_2 \rangle$ and consider the matrix

$D_{LL’} = \begin{bmatrix} M & \frac{g}{h} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} N & \frac{i}{j} \\ 0 & 1 \end{bmatrix}^{-1}$ and let $\alpha$ be the smallest positive rational number such that all entries of the matrix $\alpha.D_{LL’}$ are integers, then

$\delta(L,L’) = det(\alpha.D_{LL’}) \in \mathbb{N}$ defines a symmetric hyperdistance which depends only of the equivalence classes of lattices (hyperdistance because the log of it behaves like an ordinary distance).

Conway’s big picture is the graph obtained by taking as its vertices the equivalence classes of lattices commensurable with $L_1$ and with edges connecting any two lattices separated by a prime number hyperdistance. Here’s part of the 2-picture, that is, only depicting the edges of hyperdistance 2.

The 2-picture is an infinite 3-valent tree as there are precisely 3 classes of lattices at hyperdistance 2 from any lattice $L = \langle v_1,v_2 \rangle$ namely (the equivalence classes of) $\langle \frac{1}{2}v_1,v_2 \rangle~,~\langle v_1, \frac{1}{2} v_2 \rangle$ and $\langle \frac{1}{2}(v_1+v_2),v_2 \rangle$.

Similarly, for any prime hyperdistance p, the p-picture is an infinite p+1-valent tree and the big picture is the product over all these prime trees. That is, two lattices at square-free hyperdistance $N=p_1p_2\ldots p_k$ are two corners of a k-cell in the big picture!
(Astute readers of this blog (if such people exist…) may observe that Conway’s big picture did already appear here prominently, though in disguise. More on this another time).

The big picture presents a simple way to look at arithmetic groups and makes many facts about them visually immediate. For example, the point-stabilizer subgroup of $L_1$ clearly is the modular group $PSL_2(\mathbb{Z})$. The point-stabilizer of any other lattice is a certain conjugate of the modular group inside $PSL_2(\mathbb{R})$. For example, the stabilizer subgroup of the lattice $L_N = \langle Ne_1,e_2 \rangle$ (at hyperdistance N from $L_1$) is the subgroup

${ \begin{bmatrix} a & \frac{b}{N} \\ Nc & d \end{bmatrix}~|~\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in PSL_2(\mathbb{Z})~}$

Now the intersection of these two groups is the modular subgroup $\Gamma_0(N)$ (consisting of those modular group element whose lower left-hand entry is divisible by N). That is, the proper way to look at this arithmetic group is as the joint stabilizer of the two lattices $L_1,L_N$. The picture makes it trivial to compute the index of this subgroup.

Consider the ball $B(L_1,N)$ with center $L_1$ and hyper-radius N (on the left, the ball with hyper-radius 4). Then, it is easy to show that the modular group acts transitively on the boundary lattices (including the lattice $L_N$), whence the index $[ \Gamma : \Gamma_0(N)]$ is just the number of these boundary lattices. For N=4 the picture shows that there are exactly 6 of them. In general, it follows from our knowledge of all the p-trees the number of all lattices at hyperdistance N from $L_1$ is equal to $N \prod_{p | N}(1+ \frac{1}{p})$, in accordance with the well-known index formula for these modular subgroups!

But, there are many other applications of the big picture giving a simple interpretation for the Hecke operators, an elegant proof of the Atkin-Lehner theorem on the normalizer of $\Gamma_0(N)$ (the whimsical source of appearances of the number 24) and of Helling’s theorem characterizing maximal arithmetical groups inside $PSL_2(\mathbb{C})$ as conjugates of the normalizers of $\Gamma_0(N)$ for square-free N.
J.H. Conway’s paper “Understanding groups like $\Gamma_0(N)$” containing all this material is a must-read! Unfortunately, I do not know of an online version.

While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry group $A_5$.

The alternating group $A_5$ has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say $C$ and construct a graph with vertices the 12 elements of $C$ and an edge between two $u,v \in C$ if and only if the group-product $u.v \in C$ still belongs to the same conjugacy class.

Observe that this relation is symmetric as from $u.v = w \in C$ it follows that $v.u=u^{-1}.u.v.u = u^{-1}.w.u \in C$. The graph obtained is the icosahedron, depicted on the right with vertices written as words in two adjacent elements u and v from $C$, as indicated.

Kostant writes : “Normally it is not a common practice in group theory to consider whether or not the product of two elements in a conjugacy class is again an element in that conjugacy class. However such a consideration here turns out to be quite productive.”

Still, similar constructions have been used in other groups as well, in particular in the study of the largest sporadic group, the monster group $\mathbb{M}$.

There is one important catch. Whereas it is quite trivial to multiply two permutations and verify whether the result is among 12 given ones, for most of us mortals it is impossible to do actual calculations in the monster. So, we’d better have an alternative way to get at the icosahedral graph using only $A_5$-data that is also available for the monster group, such as its character table.

Let $G$ be any finite group and consider three of its conjugacy classes $C(i),C(j)$ and $C(k)$. For any element $w \in C(k)$ we can compute from the character table of $G$ the number of different products $u.v = w$ such that $u \in C(i)$ and $v \in C(j)$. This number is given by the formula

$\frac{|G|}{|C_G(g_i)||C_G(g_j)|} \sum_{\chi} \frac{\chi(g_i) \chi(g_j) \overline{\chi(g_k)}}{\chi(1)}$

where the sum is taken over all irreducible characters $\chi$ and where $g_i \in C(i),g_j \in C(j)$ and $g_k \in C(k)$. Note also that $|C_G(g)|$ is the number of $G$-elements commuting with $g$ and that this number is the order of $G$ divided by the number of elements in the conjugacy class of $g$.

The character table of $A_5$ is given on the left : the five columns correspond to the different conjugacy classes of elements of order resp. 1,2,3,5 and 5 and the rows are the character functions of the 5 irreducible representations of dimensions 1,3,3,4 and 5.

Let us fix the 4th conjugacy class, that is 5a, as our class $C$. By the general formula, for a fixed $w \in C$ the number of different products $u.v=w$ with $u,v \in C$ is equal to

$\frac{60}{25}(\frac{1}{1} + \frac{(\frac{1+\sqrt{5}}{2})^3}{3} + \frac{(\frac{1-\sqrt{5}}{2})^3}{3} – \frac{1}{4} + \frac{0}{5}) = \frac{60}{25}(1 + \frac{4}{3} – \frac{1}{4}) = 5$

Because for each $x \in C$ also its inverse $x^{-1} \in C$, this can be rephrased by saying that there are exactly 5 different products $w^{-1}.u \in C$, or equivalently, that the valency of every vertex $w^{-1} \in C$ in the graph is exactly 5.

That is, our graph has 12 vertices, each with exactly 5 neighbors, and with a bit of extra work one can show it to be the icosahedral graph.

For the monster group, the Atlas tells us that it has exactly 194 irreducible representations (and hence also 194 conjugacy classes). Of these conjugacy classes, the involutions (that is the elements of order 2) are of particular importance.

There are exactly 2 conjugacy classes of involutions, usually denoted 2A and 2B. Involutions in class 2A are called “Fischer-involutions”, after Bernd Fischer, because their centralizer subgroup is an extension of Fischer’s baby Monster sporadic group.

Likewise, involutions in class 2B are usually called “Conway-involutions” because their centralizer subgroup is an extension of the largest Conway sporadic group.

Let us define the monster graph to be the graph having as its vertices the Fischer-involutions and with an edge between two of them $u,v \in 2A$ if and only if their product $u.v$ is again a Fischer-involution.

Because the centralizer subgroup is $2.\mathbb{B}$, the number of vertices is equal to $97239461142009186000 = 2^4 * 3^7 * 5^3 * 7^4 * 11 * 13^2 * 29 * 41 * 59 * 71$.

From the general result recalled before we have that the valency in all vertices is equal and to determine it we have to use the character table of the monster and the formula. Fortunately GAP provides the function ClassMultiplicationCoefficient to do this without making errors.

 gap> table:=CharacterTable("M"); CharacterTable( "M" ) gap> ClassMultiplicationCoefficient(table,2,2,2); 27143910000 

Perhaps noticeable is the fact that the prime decomposition of the valency $27143910000 = 2^4 * 3^4 * 5^4 * 23 * 31 * 47$ is symmetric in the three smallest and three largest prime factors of the baby monster order.

Robert Griess proved that one can recover the monster group $\mathbb{M}$ from the monster graph as its automorphism group!

As in the case of the icosahedral graph, the number of vertices and their common valency does not determine the monster graph uniquely. To gain more insight, we would like to know more about the sizes of minimal circuits in the graph, the number of such minimal circuits going through a fixed vertex, and so on.

Such an investigation quickly leads to a careful analysis which other elements can be obtained from products $u.v$ of two Fischer involutions $u,v \in 2A$. We are in for a major surprise, first observed by John McKay:

Printing out the number of products of two Fischer-involutions giving an element in the i-th conjugacy class of the monster,
where i runs over all 194 possible classes, we get the following string of numbers :

 97239461142009186000, 27143910000, 196560, 920808, 0, 3, 1104, 4, 0, 0, 5, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 

That is, the elements of only 9 conjugacy classes can be written as products of two Fischer-involutions! These classes are :

• 1A = { 1 } written in 97239461142009186000 different ways (after all involutions have order two)
• 2A, each element of which can be written in exactly 27143910000 different ways (the valency)
• 2B, each element of which can be written in exactly 196560 different ways. Observe that this is the kissing number of the Leech lattice leading to a permutation representation of $2.Co_1$.
• 3A, each element of which can be written in exactly 920808 ways. Note that this number gives a permutation representation of the maximal monster subgroup $3.Fi_{24}’$.
• 3C, each element of which can be written in exactly 3 ways.
• 4A, each element of which can be written in exactly 1104 ways.
• 4B, each element of which can be written in exactly 4 ways.
• 5A, each element of which can be written in exactly 5 ways.
• 6A, each element of which can be written in exactly 6 ways.

Let us forget about the actual numbers for the moment and concentrate on the orders of these 9 conjugacy classes : 1,2,2,3,3,4,4,5,6. These are precisely the components of the fundamental root of the extended Dynkin diagram $\tilde{E_8}$!

This is the content of John McKay’s E(8)-observation : there should be a precise relation between the nodes of the extended Dynkin diagram and these 9 conjugacy classes in such a way that the order of the class corresponds to the component of the fundamental root. More precisely, one conjectures the following correspondence

This is similar to the classical McKay correspondence between finite subgroups of $SU(2)$ and extended Dynkin diagrams (the binary icosahedral group corresponding to extended E(8)). In that correspondence, the nodes of the Dynkin diagram correspond to irreducible representations of the group and the edges are determined by the decompositions of tensor-products with the fundamental 2-dimensional representation.

Here, however, the nodes have to correspond to conjugacy classes (rather than representations) and we have to look for another procedure to arrive at the required edges! An exciting proposal has been put forward recently by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.

It will take us a couple of posts to get there, but for now, let’s give the gist of it : monstrous moonshine gives a correspondence between conjugacy classes of the monster and certain arithmetic subgroups of $PSL_2(\mathbb{R})$ commensurable with the modular group $\Gamma = PSL_2(\mathbb{Z})$. The edges of the extended Dynkin E(8) diagram are then given by the configuration of the arithmetic groups corresponding to the indicated 9 conjugacy classes! (to be continued…)