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created: 2023-03-26T11:58
updated: 2024-05-05T11:40
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# Reflection groups

#todo/questions 

- What is the Coxeter diagram for $D_{2m} = \gens{s_1, s_2 \st s_1^2, s_2^2, (s_1s_2)^m}$, corresponding to the noncompact tiling of $\EE^2$ by $m$ congruent sectors of angle $2\pi/m$.
	- Pic: ![](../../attachments/2024-01-02.png)
	- I think this is just $(-2) \mapsvia{m} (-2)$.
- $D_{2m}$ for $m=4$ is the Coxeter group of $B_2$. Does this generalize to all $m$?
	- Answer: yes, they are associated to $I_2(m)$, but not always type $B$. Note $I_2(3) = A_2$. ![](../../attachments/2024-01-02-2.png)
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