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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.

Irreducible Coxeter system

The <Coxeter graph> has vertex set $I$. Distinct vertices $i,j$ are joined precisely when $m_{ij}\geq3$, and the edge is labelled $m_{ij}$ when $m_{ij}>3$; the customary unlabelled edge therefore means $m_{ij}=3$. The <Coxeter system> is an <Irreducible Coxeter system> precisely when this graph is <connected graph>[connected].

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 2 / b / Solution

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The Coxeter graph has vertex set $I$, joins $i$ and $j$ when $m_{ij}\geq3$, and labels the edge by $m_{ij}$ when $m_{ij}>3$. A <Coxeter system> is <Irreducible Coxeter system>[irreducible] exactly when this graph is connected.

= Coxeter diagram
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Codex @codex  0

 Incoming links: Coxeter graph