6
$\begingroup$

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, $(s_is_j)^{m_{ij}}=e$ for some $2\leq m_{ij}\leq \infty$.

I don't know why viewed as an abstract group, every reflection group is a Coxeter group? Can somebody give me an example to explain this? Thanks in advance.

$\endgroup$
  • $\begingroup$ It says on the Wikipedia page there that every reflection group is a specific example of a Coxeter group. $\endgroup$ – Nick Mar 21 '16 at 21:07
  • $\begingroup$ @Nick Please you tell me why every reflection group is a specific example of a Coxeter group. Can you give me an example? Thanks in advance. $\endgroup$ – bing Mar 22 '16 at 1:34
6
$\begingroup$

We have $(s_i)^2 = e$ because if we repeat the same reflection twice in a row we end up back where we started.

Since $s_i s_j$ is an element of a group, it has an order (possibly infinity), which we denote by $m_{ij}$.

So in any reflection group, conditions of this form are at least satisfied. It remains to show that they're sufficient to completely define the group. As far as I know this is not so easy to explain, but Coxeter does this by characterizing the possible fundamental domains of a reflection group and then exploiting their polyhedral geometry. See:

where this is Theorem 8.

$\endgroup$

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.