While Humphreys gives a classification of finite irreducible Coxeter groups by their "geometric representation", Bjorner and Brenti (Combinatorics of Coxeter Groups) leave it as an exercise in Chapter 1 (question 4). At this point, they have covered only basic definitions, the exchange condtion and the deletion condition and given a few examples. The question is as follows:
Show with a direct and elementary argument that the Coxeter diagram $D$ of a finite irreducible Coxeter group must satisfy the following graph-theoretic requirements:
- $D$ is a tree
- $D$ has at most one vertex of degree 3 and none of higher degree
- $D$ has at most one marked (i.e. label $\geq 4$) edge
- If $D$ has a degree 3 vertex, then all edges are unmarked
Hint: In the presence of any violation, exhibit an element of infinite order.
It is not so clear to me how to prove my given sequence of words is of irreducible words, or even a sequence of strictly increasing length.
For instance, consider $\tilde{A_2} = \langle s_1, s_2, s_3 \mid s_i^2 = 1, (s_is_j)^3 = 1 \rangle$ for $j \neq i$. This group's Coxeter diagram contains a cycle, and we wish to find an element of infinite order. Such an element is given by $(s_1s_2s_3)$. By the "word property" (Theorem 3.3.1) they prove two chapters later, $(s_1s_2s_3)^n$ cannot be reduced, as it contains no instance of $s_is_{i+1}s_i$ or $s_i^2$. However, this is not an elementary argument. Any insight into how to prove this with elementary arguments would be appreciated.