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I've recently started investigating abstract algebra and have now stumbled upon "Coxeter Groups", which are a mystery to me.

I've read that Coxeter Groups

  • have something to do with reflections (in which way is entirely unclear)
  • are related to "Coxeter Matrices" and -"Diagrams" (which I don't know)
  • are groups with a certain "presentation" (which I've looked into but not understood their connection to this)

My guess is that Coxeter groups are groups of reflections "generated" by some reflections - but I neither know if the terms here are used in any way correctly nor if it's right, partially since there are no useful examples to be found anywhere.

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    $\begingroup$ A good book to read is "Reflection Groups and Coxeter Groups" by Humphreys. If you don't know about presentation, you might want to read something about geometric group theory before. To clear one misconception: Not every element of a Coxeter group is a reflection, but they are indeed generated by reflections. The most instructive example of a coxeter group would be the dihedral groups, which are finite Coxeter groups. $\endgroup$
    – Babelfish
    Aug 27 '18 at 7:08
  • $\begingroup$ What would the non-reflection elements of a coxeter group represent, then? $\endgroup$ Aug 28 '18 at 10:16
  • $\begingroup$ In my example in the answer, consider $st$. It's a rotation, not a reflection. In general, the conjugates of the generators are the only reflections. $\endgroup$
    – Babelfish
    Aug 28 '18 at 10:18
  • $\begingroup$ This comes at it from a different direction, and I found it tremendously helpful. math.stackexchange.com/questions/735679/… $\endgroup$
    – MJD
    Feb 9 '19 at 12:32
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I'll try to sketch the connections between your bullet points with an easy example of a Coxeter group. You probably want to study some basic algebraic knowledge to follow this.

We examine the dihedral group of the hexagon $W=A_2$. This is a hexagon: This is a hexagon

We pick two opposed vertices and consider the reflection given by the line through those two vertices, call it $s$. Take an "adjacent" reflection, call it $t$. Now $s$ and $t$ generate a group of order $6$ (you should convice yourself that this is true). This group is a Coxeter group.

  • It has something to do with reflections.
  • The Coxeter matrix and the Coxeter diagram are a way to encode the properties of the Coxeter group. In our case, we have the Coxeter matrix $$M=\begin{pmatrix}2&3\\3&2\end{pmatrix}$$ and the Coxeter diagram $\circ \overset{3}{-} \circ$. The $2$'s in the matrix tell you that $s\circ s=t\circ t=\text{id}$ and the $3$'s in the matrix and in the diagram tell you that $(s\circ t)^3=\text{id}$.
  • This is made precise with the presentation of the Coxeter group. In our case, the presentation is $W \cong \langle s,t \mid s^2=t^2=ststst=1\rangle$. This presentation is an abstract way to define the Coxeter group: Take all finite words in the letters $s$ and $t$. This would be set contaning elements as $s$, $sts$, $tttss$, $ttssttttsst$ or even the empty word $\varepsilon$. Now we introduce an equivalence relation: We call two words equivalent, if one can be obtained from the other by deleting or inserting $ss$, $tt$ or $ststst$. Concatenation of two words gives a multiplication: $st*sst=stsst \sim stt \sim s$. It turns out that, modulo this equivalence relation, words in $s$ and $t$ are a group. And it is isomorphic to the reflection group we defined above.

To summerize: Coxeter diagram and Coxeter matrix are a tool to encode the presentation of the Coxeter group. Each Coxeter group has such a special representation. Each Coxeter group can be realized geometrically as a group generated by reflection of "something".

Edit: To answer the question how the "non-reflection" look like: The element $st\in A_2$ is not a reflection, it is a rotation by $120^\circ$(the top vertex at $0^\circ$ gets mapped to $240^\circ$, the $60^\circ$-vertex gets mapped to $300^\circ$). In general, the reflections of a Coxeter group (often denoted by $T$, in contrast to the set of generating, simple reflections $S$) are precisely the conjugates of the generators, i.e. $s$, $t$ and $sts=tst$ in our example.

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