The Coxeter graph has vertex set , joins and when , and labels the edge by when . A Coxeter system is irreducible exactly when this graph is connected.
An irreducible Coxeter system is one whose Coxeter graph is connected. The connected components of a Coxeter graph give the direct-product decomposition of its Coxeter group.
Articles by others on the same topic
The Coxeter graph is an important concept in the fields of algebra, geometry, and graph theory. Specifically, it is a particular type of graph that represents the symmetric group and the properties of certain mathematical structures, particularly in relation to Coxeter groups. Here are some key features of the Coxeter graph: 1. **Definition and Structure**: The Coxeter graph is a finite undirected graph with 12 vertices and 18 edges.