occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Thursday, July 17, 2008

Lieven's Trinities II

Lieven Le Bruyn explains some of the mysteries of the monster, which is associated to a Riemann surface of genus

$g = 9619255057077534236743570297163223297687552000000001$

That's an awful lot of gluing of heptagon edges, which when halved define the ribbon graph for the surface. Lieven's construction involves our favourite modular group and its group algebra, basically all possible combinations of elements of the group. Let's start out with the much simpler M Theory group $S_{3}$, of permutations on three letters. The group algebra is the complex number combinations, such as the 1-circulants

$a \cdot 1 + b \cdot (312) + c \cdot (231)$

or mixtures of 1-circulants and 2-circulants. These algebras showed up in the Hopf algebra triples associated to operad polytopes like the permutohedra and associahedra, as investigated by Loday et al. In a physics variant on Lieven's challenge: can you match these numbers to something concrete, like particle spectra and mixing parameters? I'll buy the winner a few pints of good South Island beer.

Aside: The Kostant of Lisi fame has posted some interesting email correspondence on his door. It is available here.