Geometric graph theory: Weyl Groups, Root Systems and Quadratic Forms

In this video we explore the geometry of a graph coming from a combinatorial game. By playing the Mutation Game on populations on a graph, i.e. integer valued functions on the vertices, we can generate special populations which form a root system. In the ADE cases these are well-studied, finite sets of vectors invariant under certain reflections. The group generated by the mutations at the various vertices maybe called a Weyl group or Coxeter group. In the special case of ADE graphs these are finite groups. We introduce a general symmetric bilinear form for which the mutations are actually reflections, giving us a Weyl/Coxeter group of isometries. This is then quite general: every graph gives us a distinguished geometrical structure on its populations, and a group of reflections. An important special case is the symmetric group associated to the A_n diagrams, and the Mutation Game gives us explicit representations of that connected with polygons generalizing the permutahedron. Thanks to Sean Gardiner for videoing, and to Joshua Capel for help setting up. Also please note that my original statement of the Mutation Game around 3:00 was not correct, but corrected later in the talk thanks to a helpful comment from the audience!