Coxeter Element - Coxeter Plane

For a given Coxeter element w, there is a unique plane P on which w acts by rotation by 2π/h. This is called the Coxeter plane and is the plane on which P has eigenvalues e2πi/h and e−2πi/h = e2πi(h−1)/h. This plane was first systematically studied in (Coxeter 1948), and subsequently used in (Steinberg 1959) to provide uniform proofs about properties of Coxeter elements.

The Coxeter plane is often used to draw diagrams of higher-dimensional polytopes and root systems – the vertices and edges of the polytope, or roots (and some edges connecting these) are orthogonally projected onto the Coxeter plane, yielding a Petrie polygon with h-fold rotational symmetry. For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under w form h-fold circular arrangements and there is an empty center, as in the E₈ diagram at above right. For polytopes, a vertex may map to zero, as depicted below. Projections onto the Coxeter plane are depicted below for the Platonic solids.

• Petrie polygons of the Platonic solids, showing 4-fold, 6-fold, and 10-fold symmetry, corresponding to the Coxeter lengths of A₃, BC₃, and H₃.