Meaning of Coxeter diagram | Babel Free

Definitions

A Coxeter-Dynkin diagram.

Examples

“It suffices to make⟳ use⟳ of the tables above and observe⟳ that the Coxeter diagram of a direct⟳ sum of C-matrices is the disconnected union of the Coxeter diagrams of the summands. For the description of C⁻-matrices satisfying condition (2) we shall use⟳ Coxeter diagrams with the additional stipulation that in case aᵢⱼ < −1 the vertices vᵢ and vⱼ are joined by a dotted line with index −aᵢⱼ. By the Coxeter diagram of a Coxeter polyhedron and a lattice generated by reflections we mean⟳ the Coxeter diagram of the corresponding C⁻-matrix.”

“An automorphism of a Coxeter diagram M leads in a natural way to a Coxeter subgroup of the Coxeter group of type⟳ M. We introduce⟳ admissible partitions of Coxeter diagrams in order⟳ to generalize this situation.”

“The Coxeter diagrams of type⟳ E₇ and E₈ have⟳ no nontrivial automorphisms, so σ₀ is trivial in those cases.[…]The Coxeter diagram of type⟳ F₄ has a unique nontrivial automorphism, but σ₀ is trivial in this case. One can see⟳ this by using the Dynkin diagram instead of the Coxeter diagram and noting that it does not have⟳ any nontrivial automorphisms.”

CEFR level