Free presentations of Weyl groups

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.).

1 Answer

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.

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