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A forgotten type and roots of unity (again)

The monstrous moonshine picture is the finite piece of Conway’s Big Picture needed to understand the 171 moonshine groups associated to conjugacy classes of the monster.

Last time I claimed that there were exactly 7 types of local behaviour, but I missed one. The forgotten type is centered at the number lattice 84.

Locally around it the moonshine picture looks like this
Misplaced &

and it involves all square roots of unity (42, 4212 and 168) and 3-rd roots of unity (28, 2813, 2823 and 252) centered at 84.

No, I’m not hallucinating, there are indeed 3 square roots of unity and 4 third roots of unity as they come in two families, depending on which of the two canonical forms to express a lattice is chosen.

In the ‘normal’ expression Mgh the two square roots are 42 and 4212 and the three third roots are 28,2813 and 2823. But in the ‘other’ expression
Mgh=(gh,1h2M)
(with g.g1 mod h) the families of 2-nd and 3-rd roots of unity are
{4212=(12,1168),168=(0,1168)}
and
{2813=(13,1252),2823=(23,1252),252=(0,1252)}
As in the tetrahedral snake post, it is best to view the four 3-rd roots of unity centered at 84 as the vertices of a tetrahedron with center of gravity at 84. Power maps in the first family correspond to rotations along the axis through 252 and power maps in the second family are rotations along the axis through 28.

In the ‘normal’ expression of lattices there’s then a total of 8 different local types, but two of them consist of just one number lattice: in 8 the local picture contains all square, 4-th and 8-th roots of unity centered at 8, and in 84 the square and 3-rd roots.

Perhaps surprisingly, if we redo everything in the ‘other’ expression (and use the other families of roots of unity), then the moonshine picture has only 7 types of local behaviour. The forgotten type 84 appears to split into two occurrences of other types (one with only square roots of unity, and one with only 3-rd roots).

I wonder what all this has to do with the action of the Bost-Connes algebra on the big picture or with Plazas’ approach to moonshine via non-commutative geometry.

Published in geometry groups math noncommutative