Can someone please provide me with a reference for the fact that the Weyl group $W$ associated to a root system $\Phi$ can be realised as a Coxeter group?

This means that a Weyl group $W$ has a presentation of the form $\langle s_i \mid s_i^2, (s_is_j)^{n_{ij}}\rangle.$

This is normally done by choosing a set of positive roots $\Phi^+$ and a set of simple roots $\Sigma=\lbrace \alpha_i \rbrace \subset \Phi^+$ and mapping the generators of the free group $\langle s_i\rangle$ to the $s_{\alpha_i}$, the reflections across the hyperplane perpendicular to each $\alpha_i$. If we choose the $n_{ij}$ to be the order of $s_{\alpha_i}s_{\alpha_j}$ then clearly this map factors to the quotient.

The difficulty is in showing that the map is injective. If we happen to have $s_{i_1}\dots s_{i_n}$ and $s_{j_i}\dots s_{j_m}$ mapping to the same element $w$, we must show that the former elements are the same in the quotient of the free group. I have a vague understanding that this can be done by looking at the two induced 'galleries' connecting the fundamental domain (Weyl chamber)) $C$ with $wC$.

I hope someone can provide a reference with all the details of the argument in the preceeding paragraph (preferably not Bump's Lie Groups 2nd Ed.).


A proof can be found in Section 1.9 of James E. Humphreys’ Reflection groups and Coxeter groups. However, this proof does not use Weyl chambers but instead argues that any relation $s_{\alpha_{i_1}} \dotsm s_{\alpha_{i_n}} = 1$ in $W$ can be deduced from the given relations $(s_{\alpha_i} s_{\alpha_j})^{n_{ij}} = 1$.

(Disclaimer: I’ve never read through the proof myself, so I can’t say how well this works.)

I think that another source is Bourbaki’s Lie Groups and Lie Algebras, Chapter VI, §1, no. 5, Theorem 2, part (vii) (page 166 in my edition). I seems that (in true Bourbaki fashion) one has to follow the rabbit hole given references to the earlier chapter to assemble the complete proof. But at least a quick glance at the first reference (Chapter V, §3, no. 2, Theorem 1) suggests that this proof uses Weyl chambers.

  • $\begingroup$ Thanks! Will take a look at these proofs. $\endgroup$ – BetaTumSeNaHoPaega Jun 15 at 15:30

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