It’s always tricky to blog about work in progress.I’ve taken some time to turn my sketchy notes on the the moonshine picture into a more formal, or at least a checkable, write-up.
It is still a draft version 0.1 but in case you are interested to debug it, here’s the monstrous moonshine picture (pdf).
No doubt there are typos (I didn’t even pull it through Excalibur) and errors left. Comments are welcome.
In the monster dictates her picture I claimed to have an elaborate procedure to recover the moonshine picture from group-theoretical monster data.
Conjugacy classes 24J and 8C of the monster appear to be somewhat special when it comes to moonshine. Using their power-up classes (as given in the Atlas Monster page) one can determine the threads of the moonshine groups corresponding to all conjugacy classes, except for 14 exceptions:
These exceptional conjugacy classes turn out to be exactly the power-ups of conjugacy classes 8B,8D,16A and 16C. This seems to give a purely group-theoretical method to determine for each conjugacy class at least the thread of the corresponding moonshine group.
No idea whether this is known to the experts. It seems to indicate that a small set of conjugacy classes are responsible for most of moonshine.
Next up my agenda, to fit in the Atkin-Lehner involutions and the connection with non-commutative geometry, which was the original motivation to collect this data.