Week Two Friday: Reflections and Coxeter Groups

Today, we had an exciting guest lecture by Dr. Kasia Jankiewicz, a professor at the University of California, Santa Cruz. Dr. Jankiewicz discussed reflections, which give us a geometric interpretation for elements of certain groups. For instance, (at least) half of the elements of the dihedral group of order $D_n$ can be interpreted as reflections of the regular $n$ -gon. Below is a summary of the topics Dr. Jankiewicz discussed.

Consider a reflection $r$ across a line $L$ in $\mathbb{R}^2$ :

Such a reflection brings a point $p \in \mathbb{R}^2$ to a point $r(p)$ on the other side of $L$ .

Similarly, in $\mathbb{R}^3$ we can consider a reflection over a plane $P$ :

More generally, in $\mathbb{R}^n$ there exists a reflection across any copy of $\mathbb{R}^{n-1}$ . Our goal for today is to look at groups generated by reflections.

Consider a regular pentagon on the plane. Let $r$ and $s$ be reflections across the lines $L_r$ and $L_s$ which intersect at angle $\pi / 5$ .

How do each of these reflections permute the vertices $\{ v_1,v_2,v_3,v_4,v_5\}$ ?

$r: v_1$ , $v_2$ ⟷ $v_5$ , $v_3$ ⟷ $v_4$ .

$s: v_3$ , $v_2$ ⟷ $v_4$ , $v_1$ ⟷ $v_5$ .

What does $rs$ do?

$rs: v_1$ ⟶ $v_2$ ⟶ $v_3$ ⟶ $v_4$ ⟶ $v_5$ ⟶ $v_1$ …

Therefore, $rs$ is a rotation by $\frac{2\pi}{5}$ about the center of the pentagon. Because we know $r$ and $rs$ generate $D_{10}$ , we can conclude that $r$ and $s$ generate $D_{10}$ as well, giving us the the group presentation:

$D_{10} = \langle r,s | r^2 = s^2 = (rs)^5 = e \rangle$

We can generalize the above conclusions. If $r$ and $s$ are reflections about lines intersecting at the angle $\frac{\pi}{n}$ , then $rs$ is a rotation about the intersecting point by angle $\frac{2\pi}{n}$ .

Let’s look at another group generated by reflections. Consider the real line. Let $r$ denote the reflection across $x = 0.5$ , and $s$ denote the reflection across $x = -0.5$ .

How do these reflections transform points on the line?

$r: 0$ ⟷ $1$ , $-1$ ⟷ $2$ , $-2$ ⟷ $3$ ,…

$s: 0$ ⟷ $-1$ , $1$ ⟷ $-2$ , $2$ ⟷ $-3$ ,…

In general, $r(x) = 1-x$ and $s(x) = -1-x$ . Composing $r$ and $s$ yields a translation across the real line:

$rs(x) = 1-(-1-x) = x+2$

In general, if $r$ and $s$ are reflections about non-intersecting lines $L_r$ and $L_s$ at distance $k$ from each other, then $rs$ is a translation by distance $2k$ orthogonal to those lines. We call the group generated by two such reflections the infinite dihedral group, denoted

$D_{\infty} = \langle r,s | r^2 = s^2 = e \rangle$

Note that the direction in which $rs$ shifts points on the line depends on the relative positions of $L_r$ and $L_s$ . If $L_r$ lies to the right of $L_s$ , then $rs(x)$ lies to the right of $x$ .

Now, let’s look at an example of a group generated by reflections in $\mathbb{R}^3$ . Consider the cube with corners $(\pm 1, \pm 1, \pm 1)$ . Let $r$ , $s$ , and $t$ be reflections across the planes $x = 0$ , $y = 0$ , and $z = 0$ respectively.

The planes $x = 0$ and $y = 0$ intersect at angle $\pi/2$ , so $rs$ is a rotation about the $x$ -axis by angle $\pi$ . This yields the following presentation of the group of symmetries of a cube:

$\langle r, s, t | r^2 = s^2 = t^2 = (rs)^2 = (st)^2 = (tr)^2 \rangle$

Note that, given the above abstract presentation alone, you may not be able to distinguish between generators that change the orientation of the cube and ones that preserve it.

Now, let’s consider another infinite group generated by reflections. Consider an equilateral triangle in $\mathbb{R}^2$ and reflections about the lines bordering its edges. We’ll say that three edges corresponding to the reflections $r$ , $s$ , and $t$ are colored red, blue, and green respectively. In the drawing below, the red, green, and blue edges lie along red, green, and blue lines:

The angle between the red and blue lines is $\pi / 3$ , so $(rs)^3 = e$ . When we perform a reflection along one edge, observe that the lines along which the other edges lie are rotated. The transformations of the equilateral triangle yielded by flipping along edges can form a group. For instance, if we consider the group generated by flipping across the edges $r$ and $s$ , we get six triangles forming a hexagon:

However, the group generated by $r$ , $s$ , and $t$ together gives a tessellation of the plane with equilateral triangles, yielding the following presentation:

$\langle r, s, t | r^2 = s^2 = t^2 = (rs)^3 = (st)^3 = (tr)^3 \rangle$

For a more detailed treatment of the above group, see chapter 2 of John Meier’s Groups, Graphs, and Trees. This group is an example of a triangle group, a group generated by reflections of a triangle across its edges. It is also an example of a Coxeter group!

Definition: A Coxeter group $W$ is a group with a presentation of the form:

$W = \langle s_1, s_2, ..., s_n | s_i^2, (s_i s_j)^{m_{i,j}} \rangle$

where $m_{i,j} = m_{j,i} \in \{ 2, 3, ...\} \cup \{ \infty \}$ . If we say $m_{i,j} = \infty$ , we mean that there is no relation of the form $(s_i s_j)^k$ for $k \in \mathbb{R}$ .

If you’re interested in learning more about the abstract geometric constructions described by Coxeter groups, see buildings, Coxeter complexes, the Davis complex, or Tits representations.

Here are a couple more neat facts that we briefly touched on regarding Coxeter groups:

All the data in a Coxeter group can be encoded in a labelled graph, whose vertices correspond to generators and whose edges are labelled by the finite $m_{i,j}$ s. In other words, if $(s_i s_j)^{m_{i,j}}$ is a relation in the presentation of a Coxeter group, there is an edge of weight $m_{i,j}$ between $s_i$ and $s_j$ . Note that, because $m_{i,j} = m_{j,i}$ for all $i,j$ , we don’t need to orient the edges of this graph.
Coxeter groups are linear, meaning any Coxeter group is isomorphic to a group of invertible matrices under matrix multiplication. It is not immediately apparent to me why this is the case, but it seems pretty neat.