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The Big Picture is non-commutative

Conwayโ€™s Big Picture consists of all pairs of rational numbers M,gh with M>0 and 0โ‰คgh<1 with (g,h)=1. Recall from last time that M,gh stands for the lattice
Z(Meโ†’1+gheโ†’2)โŠ•Zeโ†’2โŠ‚Q2
and we associate to it the rational 2ร—2 matrix
ฮฑM,gh=[Mgh01]

If M is a natural number we write Mgh and call the corresponding lattice number-like, if g=0 we drop the zero and write M.

The Big Picture carries a wealth of structures. Today, we will see that it can be factored as the product of Bruhat-Tits buildings for GL2(Qp), over all prime numbers p.

Hereโ€™s the factor-building for p=2, which is a 3-valent tree:

To see this, define the distance between lattices to be
d(M,gh | N,ij)=log Det(q(ฮฑM,gh.ฮฑN,ijโˆ’1))
where q is the smallest strictly positive rational number such that q(ฮฑM,gh.ฮฑN,ijโˆ’1)โˆˆGL2(Z).

We turn the Big Picture into a (coloured) graph by drawing an edge (of colour p, for p a prime number) between any two lattices distanced by log(p).

Misplaced &

The p-coloured subgraph is p+1-valent.

The p-neighbours of the lattice 1=Zeโ†’1โŠ•Zeโ†’2 are precisely these p+1 lattices:

pand1p,kpfor0โ‰คk<p And, multiplying the corresponding matrices with ฮฑM,gh tells us that the p-neighbours of M,gh are then these p+1 lattices: pM,pgh mod 1andMp,1p(gh+k) mod 1for0โ‰คk<p Here's part of the 2-coloured neighbourhood of 1

To check that the p-coloured subgraph is indeed the Bruhat-Tits building of GL2(Qp) it remains to see that it is a tree.

For this it is best to introduce p+1 operators on lattices

pโˆ—andkpโˆ—for0โ‰คk<p defined by left-multiplying ฮฑM,gh by the matrices [p001]and[1pkp01]for0โ‰คk<p The lattice pโˆ—M,gh lies closer to 1 than M,gh (unless M,gh=M is a number) whereas the lattices kpโˆ—M,gh lie further, so it suffices to show that the p operators 0pโˆ—, 1pโˆ—, โ€ฆ ,pโˆ’1pโˆ— form a free non-commutative monoid.
This follows from the fact that the operator
(knpโˆ—)โˆ˜โ‹ฏโˆ˜(k2pโˆ—)โˆ˜(k1pโˆ—)
is given by left-multiplication with the matrix
[1pnk1pn+k2pnโˆ’1+โ‹ฏ+knp01]
which determines the order in which the ki occur.

A lattice at distance nlog(p) from 1 can be uniquely written as
(knโˆ’lpโˆ—)โˆ˜โ‹ฏโˆ˜(kl+1pโˆ—)โˆ˜(plโˆ—)1
which gives us the unique path to it from 1.

The Big Picture itself is then the product of these Bruhat-Tits trees over all prime numbers p. Decomposing the distance from M,gh to 1 as
d(M,gh | 1)=n1log(p1)+โ‹ฏ+nklog(pk)
will then allow us to find minimal paths from 1 to M,gh.

But we should be careful in drawing 2-dimensional cells (or higher dimensional ones) in this โ€˜productโ€™ of trees as the operators
kpโˆ—andlqโˆ—
for different primes p and q do not commute, in general. The composition
(kpโˆ—)โˆ˜(lqโˆ—)with matrix[1pqkq+lpq01]
has as numerator in the upper-right corner 0โ‰คkq+l<pq and this number can be uniquely(!) written as kq+l=up+vwith0โ‰คu<q, 0โ‰คv<p That is, there are unique operators uqโˆ— and vpโˆ— such that (kpโˆ—)โˆ˜(lqโˆ—)=(uqโˆ—)โˆ˜(vpโˆ—) which determine the 2-cells Misplaced & These give us the commutation relations between the free monoids of operators corresponding to different primes.
For the primes 2 and 3, relevant in the description of the Moonshine Picture, the commutation relations are

(02โˆ—)โˆ˜(03โˆ—)=(03โˆ—)โˆ˜(02โˆ—),(02โˆ—)โˆ˜(13โˆ—)=(03โˆ—)โˆ˜(12โˆ—),(02โˆ—)โˆ˜(23โˆ—)=(13โˆ—)โˆ˜(02โˆ—)

(12โˆ—)โˆ˜(03โˆ—)=(13โˆ—)โˆ˜(12โˆ—),(12โˆ—)โˆ˜(13โˆ—)=(23โˆ—)โˆ˜(02โˆ—),(12โˆ—)โˆ˜(23โˆ—)=(23โˆ—)โˆ˜(12โˆ—)

Published in groups math noncommutative