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The Arithmetic of Right-Angled Coxeter Groups

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by Andy Miller, University of Oklahoma

  • Colloquium
When Fri, Feb 09, 2018
from 04:10 PM to 05:00 PM
Where 202 Ritter Hall
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Each Coxeter group G with n generators has an n-dimensional representation which is orthogonal with respect to a canonical bilinear form B(x,y). The associated quadratic form B(x,x) will be Diophantine when G is right-angled. By construction, for each
integer D, the Coxeter group G acts on the set Sol(D) of primitive integer solutions to B(x,x)=D and this set admits a partial ordering whose components coincide with the orbits of the G action.

All of these constructs can be directly defined simply in terms
of the Coxeter graph associated with G. The talk will focus on describing some low dimensional cases of this construction. Examples tend to have interesting geometric interpretations involving such concepts as Apollonian circle packings, Pythagorean triples, and Conway's pictorial realizations of binary forms and their rivers.

circle packing

Tea is available at 3:30pm in the lobby of Ritter Hall, with the talk to follow at 4:10pm in 202 Ritter Hall.

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