Convex Hull of Coxeter Groups

For a given Coxeter group W(S) acting on Rⁿ, we can represent its elements as a set of vectors in the matrix space M_n(R). This project focuses on the enumeration and analysis of the facets of the resulting convex polytope.

Research Context

Facets of Coxeter polytopes have been a subject of significant interest in combinatorial geometry:

A_n and B_n: Research has proven that these groups yield only one orbit of facets, defined by the Birkhoff tensors.
F₄: While previously claimed to have only one orbit, our investigation found this to be false.
H₄: Our enumeration identified 1,063 distinct orbits of facets. Detailed results are available in this document.

References

Original Research Paper on Birkhoff tensors and Coxeter polytopes.