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Category: groups

A tetrahedral snake

A tetrahedral snake, sometimes called a Steinhaus snake, is a collection of tetrahedra, linked face to face.

Steinhaus showed in 1956 that the last tetrahedron in the snake can never be a translation of the first one. This is a consequence of the fact that the group generated by the four reflexions in the faces of a tetrahedron form the free product C2C2C2C2.

For a proof of this, see Stan Wagon’s book The Banach-Tarski paradox, starting at page 68.

The tetrahedral snake we will look at here is a snake in the Big Picture which we need to determine the moonshine group (3|3) corresponding to conjugacy class 3C of the Monster.

The thread (3|3) is the spine of the (9|1)-snake which involves the following lattices
Misplaced &
It is best to look at the four extremal lattices as the vertices of a tetrahedron with the lattice 3 corresponding to its point of gravity.

The congruence subgroup Γ0(9) fixes each of these lattices, and the arithmetic group Γ0(3|3) is the conjugate of Γ0(1)
Γ0(3|3)={[13001].[abcd].[3001]=[ab33c1] | adbc=1}
We know that Γ0(3|3) normalizes the subgroup Γ0(9) and we need to find the moonshine group (3|3) which should have index 3 in Γ0(3|3) and contain Γ0(9).

So, it is natural to consider the finite group A=Γ0(3|3)/Γ9(0) which is generated by the co-sets of
x=[11301]andy=[1030]
To determine this group we look at the action of it on the lattices in the (9|1)-snake. It will fix the central lattice 3 but will move the other lattices.

Recall that it is best to associate to the lattice M.gh the matrix
αM,gh=[Mgh01]
and then the action is given by right-multiplication.

[1001].x=[11301],[11301].x=[12301],[12301].x=[1001]
That is, x corresponds to a 3-cycle 11131231 and fixes the lattice 9 (so is rotation around the axis through the vertex 9).

To compute the action of y it is best to use an alternative description of the lattice, replacing the roles of the base-vectors e1 and e2. These latices are projectively equivalent
Z(Me1+ghe2)Ze2andZe1Z(ghe1+1h2Me2)
where g.g 1 (mod h). So, we have equivalent descriptions of the lattices
M,gh=(gh,1h2M)andM,0=(0,1M)
and we associate to the lattice in the second normal form the matrix
βM,gh=[10gh1h2M]
and then the action is again given by right-multiplication.

In the tetrahedral example we have
1=(0,13),113=(13,19),123=(23,19),9=(0,19)
and
[101319].y=[102319],[102319].y=[10019],[10019].y=[101319]
That is, y corresponds to the 3-cycle 91131239 and fixes the lattice 1 so is a rotation around the axis through 1.

Clearly, these two rotations generate the full rotation-symmetry group of the tetrahedron
Γ0(3|3)/Γ0(9)A4
which has a unique subgroup of index 3 generated by the reflexions (rotations with angle 180o around axis through midpoints of edges), generated by x.y and y.x.

The moonshine group (3|3) is therefore the subgroup generated by
(3|3)=Γ0(9),[21331],[11332]

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the 171 moonshine groups

Monstrous moonshine associates to every element of order n of the monster group M an arithmetic group of the form
(n|h)+e,f,
where h is a divisor of 24 and of n and where e,f, are divisors of nh coprime with its quotient.

In snakes, spines, and all that we’ve constructed the arithmetic group
Γ0(n|h)+e,f,
which normalizes Γ0(N) for N=h.n. If h=1 then this group is the moonshine group (n|h)+e,f,, but for h>1 the moonshine group is a specific subgroup of index h in Γ0(n|h)+e,f,.

I’m sure one can describe this subgroup explicitly in each case by analysing the action of the finite group (Γ0(n|h)+e,f,)/Γ0(N) on the (N|1)-snake. Some examples were worked out by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.

But at the moment I don’t understand the general construction given by Conway, McKay and Sebbar in On the discrete groups of moonshine. I’m stuck at the last sentence of (2) in section 3. Nothing a copy of Charles Ferenbaugh Ph. D. thesis cannot fix.

The correspondence between the conjugacy classes of the Monster and these arithmetic groups takes up 3 pages in Conway & Norton’s Monstrous Moonshine. Here’s the beginning of it.

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Snakes, spines, threads and all that

Conway introduced his Big Picture to make it easier to understand and name the groups appearing in Monstrous Moonshine.

For MQ+ and 0gh<1, M,gh denotes (the projective equivalence class of) the lattice Z(Me1+ghe2)Ze2 which we also like to represent by the 2×2 matrix αM,gh=[Mgh01] A subgroup G of GL2(Q) is said to fix M,gh if
αM,gh.G.αM,gh1SL2(Z)
The full group of all elements fixing M,gh is the conjugate
αM,gh1.SL2(Z).αM,gh
For a number lattice N=N,0 the elements of this group are all of the form
[abNcNd]with[abcd]SL2(Z)
and the intersection with SL2(Z) (which is the group of all elements fixing the lattice 1=1,0) is the congruence subgroup
Γ0(N)={[abcNd] | adNbc=1}
Conway argues that this is the real way to think of Γ0(N), as the joint stabilizer of the two lattices N and 1!

The defining definition of 24 tells us that Γ0(N) fixes more lattices. In fact, it fixes exactly the latices Mgh such that
1 | M | Nh2withh2 | Nandh | 24
Conway calls the sub-graph of the Big Picture on these lattices the snake of (N|1).

Here’s the (60|1)-snake (note that 60=22.3.5 so h=1 or h=2 and edges corresponding to the prime 2 are coloured red, those for 3 green and for 5 blue).

Misplaced &

The sub-graph of lattices fixed by Γ0(N) for h=1, that is all number-lattices M=M,0 for M a divisor of N is called the thread of (N|1). Here’s the (60|1)-thread

Misplaced &

If N factors as N=p1e1p2e2pkek then the (N|1)-thread is the product of the (piei|1)-threads and has a symmetry group of order 2k.

It is generated by k involutions, each one the reflexion in one (piei|1)-thread and the identity on the other (pjej|1)-threads.
In the (60|1)-thread these are the reflexions in the three mirrors of the figure.

So, there is one involution for every divisor e of N such that (e,Ne)=1. For such an e there are matrices, with a,b,c,dZ, of the form
We=[aebcNde]withade2bcN=e
Think of Bezout and use that (e,Ne)=1.

Such We normalizes Γ0(N), that is, for any AΓ0(N) we have that We.A.We1Γ0(N). Also, the determinant of Wee is equal to e2 so we can write We2=eA for some AΓ0(N).

That is, the transformation We (left-multiplication) sends any lattice in the thread or snake of (N|1) to another such lattice (up to projective equivalence) and if we apply We2 if fixes each such lattice (again, up to projective equivalence), so it is the desired reflexion corresponding with e.

Consider the subgroup of GL2(Q) generated by Γ0(N) and some of these matrices We,Wf, and denote by Γ0(N)+e,f, the quotient modulo positive scalar matrices, then
Γ0(N)is a normal subgroup ofΓ0(N)+e,f,
with quotient isomorphic to some (Z/2Z)l isomorphic to the subgroup generated by the involutions corresponding to e,f,.

More generally, consider the (n|h)-thread for number lattices n=n,0 and h=h,0 such that h|n as the sub-graph on all number lattices l=l,0 such that h|l|n. If we denote with Γ0(n|h) the point-wise stabilizer of n and h, then we have that
Γ(n|h)=[h001]1.Γ0(nh).[h001]
and we can then denote with
Γ0(n|h)+e,f,
the conjugate of the corresponding group Γ0(nh)+e,f,.

If h is the largest divisor of 24 such that h2 divides N, then Conway calls the spine of the (N|1)-snake the subgraph on all lattices of the snake whose distance from its periphery is exactly log(h).

For N=60, h=2 and so the spine of the (60|1)-snake is the central piece connected with double black edges

Misplaced &

which is the (30|2)-thread.

The upshot of all this is to have a visual proof of the Atkin-Lehner theorem which says that the full normalizer of Γ0(N) is the group Γ0(Nh|h)+ (that is, adding all involutions) where h is the largest divisor of 24 for which h2|N.

Any element of this normalizer must take every lattice in the (N|1)-snake fixed by Γ0(N) to another such lattice. Thus it follows that it must take the snake to itself.
Conversely, an element that takes the snake to itself must conjugate into itself the group of all matrices that fix every point of the snake, that is to say, must normalize Γ0(N).

But the elements that take the snake to itself are precisely those that take the spine to itself, and since this spine is just the (Nh|h)-thread, this group is just Γ0(Nh|h)+.

Reference: J.H. Conway, “Understanding groups like Γ0(N)”, in “Groups, Difference Sets, and the Monster”, Walter de Gruyter-Berlin-New York, 1996

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