From: mckay@cs.concordia.ca (MCKAY john) Subject: The Monster (Was: The exceptionals) Date: 20 Sep 2000 11:20:29 GMT Newsgroups: sci.physics.research Summary: [missing] In article <39C7F949.C7E10BFE@xs4all.nl>, Gerard Westendorp wrote: >baez@galaxy.ucr.edu wrote, concerning even self-dual lattices >in 24 dimensions: >> Of these, the strangest and most beautiful is the Leech >> lattice. >Does that have a Dynkin diagram? Dynkin diagrams give a description of the roots. The Leech lattice is characterized by having no roots. Let G[6,6,6] denote the (infinite) reflection group defined by the 16 node graph consisting of a tripod of three equal length legs, a,b,c each with 6 nodes with a common single central node, z = a1 = b1 = c1. Adjoining the single "spider relation" (a1a2a3b1b2b3c1c2c3)^10 = 1 to the G[6,6,6] presentation yields a presentation for the Monster wreath 2. [This is two copies of the Monster, exchanged by an element of order two.] This is a BIG finite group, far larger than the number of particles in the universe (however you care to count them!). The Monster itself is presented by a subdiagram. It is the symmetry group of a rational conformal field theory. Ref: Groups, Combinatorics, and Geometry, edited by Liebeck & Saxl. London Mathematical Society Lecture Notes vol. 165, (1992). -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.