Visualizing conjugation in affine Coxeter groups

Abstract

Affine Coxeter groups are fundamental objects in mathematics and in crystallography. If two group elements are conjugate, then they have very similar algebraic and geometric properties. Using recent structural results of Milićević, Schwer and the second author, we develop an app to visualize conjugation in affine Coxeter groups in dimensions 2 and 3. We explain the mathematics underlying this visualization, and connect its visual features to the arts.

arXiv comment13 pages, 5 figures, all in color. Version 2: added discussion of components of conjugacy classes, revised color treatment for coconjugation sets, expanded discussion of color choices for our app, added discussion of connections to the arts