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!

  • 1
    $\begingroup$ The relations given in the Wikipedia article give a presentation. $\endgroup$ – Qiaochu Yuan Nov 6 '17 at 8:43
  • $\begingroup$ Ok so there are no other relations outside those given in the article? $\endgroup$ – Dac0 Nov 6 '17 at 15:19
  • $\begingroup$ 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. $\endgroup$ – Qiaochu Yuan Nov 6 '17 at 19:35
  • $\begingroup$ No problem, Burde gave me the answer I was looking for. Thank you anyway $\endgroup$ – Dac0 Nov 7 '17 at 6:25

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.