Eigenvalues of Coxeter Elements

This lecture covers the eigenvalues of a Coxeter element and the uniqueness of the Coxeter plane associated with it. It explains how any permutation among the Coxeter generators can be achieved through cyclic permutations and transpositions. The lecture also discusses the invariance of the Coxeter number and the absence of eigenvalue 1 for a Coxeter element. Additionally, it explores the relationship between the Coxeter element and its matrix, as well as the decomposition of eigenspaces. The properties of Coxeter elements, their actions as rotations in unique planes, and the projection of root systems onto the Coxeter plane are also examined.