Geometric Graph Theory - Weyl Groups, Root Systems and Quadratic Forms

Explore the geometry of graphs derived from combinatorial games in this 50-minute lecture on Weyl Groups, Root Systems, and Quadratic Forms. Delve into the Mutation Game on graph populations, generating root systems and examining ADE cases with finite vector sets invariant under reflections. Investigate Weyl and Coxeter groups formed by vertex mutations, and discover a general symmetric bilinear form that transforms mutations into reflections. Learn how every graph yields a unique geometrical structure on its populations and a group of reflections, with special focus on the symmetric group associated with A_n diagrams. Gain insights into explicit representations of the Mutation Game connected to polygon generalizations of the permutahedron.

Introduction
Root populations
Mutations
Verification
Representation
Polytope
Polytopes
Geometry
Eigenvalues
Root systems
Conclusion