Coxeter Plane

For a finite Coxeter group acting on a Euclidean space, a Coxeter plane is a plane invariant under a Coxeter element, i.e., a product of the simple generators in some order. In the irreducible case, meaning that the Coxeter-Dynkin diagram is connected, the Coxeter element acts on this plane as a rotation through an angle 2pi/h, where h is the Coxeter number.

Orthogonal projection of a root system or polytope onto a Coxeter plane is often used to display the corresponding h-fold rotational symmetry. Such projections are related in spirit to projections in which a Petrie polygon becomes a regular polygon.

See also

Coxeter-Dynkin Diagram, Coxeter Element, Coxeter Group, Coxeter Number, Petrie Polygon, Root System

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References

Humphreys, J. E. Reflection Groups and Coxeter Groups. Cambridge, England: Cambridge University Press, 1990.Stembridge, J. R. "Coxeter Planes." https://websites.umich.edu/~jrs/coxplane.html.

Cite this as:

Weisstein, Eric W. "Coxeter Plane." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoxeterPlane.html

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