# Generators Relations of the Weyl Group

Suppose we have a Root System and call $s_{\alpha}$ the transformation $$s_{\alpha}\left(\beta\right)=\beta-\frac{2\left\langle \beta,\,\alpha\right\rangle }{\left\langle \alpha,\,\alpha\right\rangle }\alpha.$$ These transformations form a group called Weyl Group. What can I say about the presentation (i.e. generators and relations) of the Weyl Group associated to the root system of a Semi-simple Lie Algebra? Are the relations given in the Wikipedia page the only relations or there are others? Thank you in advance!

• The relations given in the Wikipedia article give a presentation. – Qiaochu Yuan Nov 6 '17 at 8:43
• Ok so there are no other relations outside those given in the article? – Dac0 Nov 6 '17 at 15:19
• That's ambiguous. Of course other relations hold; what it means for some generators and relations to give a presentation of the group is that every relation is deducible from the given relations. – Qiaochu Yuan Nov 6 '17 at 19:35
• No problem, Burde gave me the answer I was looking for. Thank you anyway – Dac0 Nov 7 '17 at 6:25

## 1 Answer

In general, there may be several presentations of a group - see K.Conrad's article on Generating Sets. A good example for this is the symmetric group. See the table after Example $1.8$ for the different possibilities. Actually, Weyl groups usually are given in the Coxeter presentation, but there are others.

• Thank you, the second reference you gave is exactly what I needed. – Dac0 Nov 6 '17 at 15:22
• You are welcome:) – Dietrich Burde Nov 6 '17 at 15:41