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``Coxeter Diagram'' for a complex hyperbolic reflection group and the bimonster

Algebra & Discrete Mathematics

Speaker: Tathagata Basak, UC Berkeley
Location: 1147 MSB
Start time: Thu, Feb 16 2006, 12:10PM

Let D be the incidence graph of the projective plane over the finite field with 3 elements. Conway, Ivanov gave a remarkably simple presentation of the wreath product of the monster with Z/2 on the Coxeter diagram D.

Let L be the direct sum of the complex Leech lattice and a hyperbolic cell.

We will describe 26 complex reflections generating the automorphism group of L that form the same Coxeter diagram D under braiding and commuting relations.

We'll see that our example has surprising analogies with the theory of Weyl group that make our proofs work. D acts as the Coxeter-Dynkin diagram for the reflection group of L. There is a parallel story for the quaternionic Leech lattice where the surprises repeat.