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!
2026-03-27 01:45:10.1774575910
Generators Relations of the Weyl Group
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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.