---
title: "A Game of Permutations, Part 3"
description: |
  Extending swap diagrams (with apologies to H.S.M. Coxeter).
format:
  html:
    html-math-method: katex
date: "2022-05-20"
date-modified: "2025-07-06"
categories:
  - graph theory
  - group theory
---

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This post assumes you have read the [first post](../1), which talks about swap diagrams
  and the symmetric group.
The section about Cayley graphs from the [second post](../2) will also be useful.

To summarize from the first post, a swap diagram is a graph where each vertex represents
  a list index and each edge represents the permutation which swaps the two entries.
I singled out three graph families for their symmetry:

- Paths, which link adjacent swaps
- Stars, in which all swaps contain a distinguished index
- Complete graphs, which contain all possible swaps

Swap diagrams are ultimately limited by only being able to describe collections of 2-cycles.
If we wanted to include higher-order elements, we'd require a weaker
  mathematical structure called a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph).
These are even more difficult to draw than simple graphs, much less understand.

Another way we are limited is in the kinds of groups which can be encoded.
One can rather quickly notice that swap diagrams will only generate symmetric groups
  and direct products thereof.
Also, while edges are members of a generating set, the vertices contain information
  which is ultimately irrelevant to the group.


Upgrading Diagrams
------------------

What if we forgot the roles of the vertices, allowed the edges to take their place?
In other words, we want to make a graph where the vertices represent generators,
  rather than the edges.

Ideally, we'd still like to preserve some information about coincident edges.
Graph-theoretically, we can elevate the role of the edges to that of the vertices
  by making a [line graph](https://en.wikipedia.org/wiki/Line_graph).
This operation produces a graph such that

- Every edge in the original graph becomes a vertex in the new graph
- Coincident edges in the new graph are connected by an edge in the new graph
    - In other words, we convert adjacent edges to adjacent vertices.

![
  While line graphs of paths simply shrink,
    $\bigstar_5$ shrinks to $K_4$ before expanding to $K_{2,2,2}$
](./line_graphs.png)

The line graph of a swap diagram results in a graph where the vertices are the 2-cycles.
In a swap diagram, when two distinct 2-cycles share an index (vertex), their product has order 3.
Thus, in the line graph, the edges mean that the product of the two vertices has order 3.
Non-adjacent vertices still have a product with order 2, since they represent disjoint 2-cycles.

To push the group/graph correspondence the even further, we can impose a final restriction.
Namely, a subgraph should represent a sub*group*.
Specifically, we want an *induced* subgraph
  (one where if two connected vertices are included, then so is the edge joining them)
  since this maintains the relationship between the generators.
Unfortunately, this restriction disqualifies many of the line graphs of swap diagrams.
At least our beloved paths are left untouched, since $L(P_n) = P_{n-1}$.


### Coxeter Diagrams

As it turns out, these restrictions are significant and produce some very deep mathematical objects,
  known as *Coxeter diagrams* (named for the same Coxeter as in [Goldberg-Coxeter](/posts/math/pentagons/1)).
In this domain, the aforementioned rules about vertices and edges apply:
  each vertex corresponds to an order 2 element and each edge signifies
  that the product of two elements has order 3.
Furthermore, *disconnected* vertices correspond to elements which commute with one another,
  just like disjoint 2-cycles.
However, each vertex need not correspond to a single 2-cycle,
  and can instead be *any* element of order 2, i.e., a product of disjoint 2-cycles.


<!-- TODO: better wikimedia imports -->
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
  <a title="Rgugliel, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File:Finite_coxeter.svg#/media/File:Finite_coxeter.svg">
    <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Finite_coxeter.svg/600px-Finite_coxeter.svg.png" style="background-color:white !important; border-radius: 5px; padding: 5px;" alt="Finite coxeter.svg">
  </a>

  <figcaption>
    The most important Coxeter diagrams.
    Note that the $A_n$ diagrams are just the familiar path graphs.
    <br>
    Source: Rgugliel, via Wikimedia Commons
  </figcaption>
</figure>
</div>

Coxeter diagrams are frequently extended to enable describing higher-order products
  by labelling the edge[^1].
For example, the product between $(1 ~ 2)(3 ~ 4)$ and $(2 ~ 3)$ has order 4.
A diagram for these elements would therefore include two vertices connected
  with an edge labelled "4", corresponding to $B_2$.

[^1]: Unlabelled edges still have order 3.


### How Big and What Shape?

We already know how big the group a swap diagram generates.
Each connected component with *n* vertices generates the group $S_n$.
They combine via the direct product to give a group of order

$$
\left | \prod_i S_{n_i} \right | = \prod_i | S_{n_i} | = \prod_i n_i!
$$

Removing an edge from a swap diagram has the potential to make the graph disconnected.
In this case, the graph is split into two parts, which map to $S_m$ and $S_n$.
This tells us that $S_m \times S_n$ embeds in $S_{m + n}$.

The subgroup restriction makes Coxeter diagrams a lot more sensitive to changes.
We at least know that if we remove a vertex, the resulting diagram is a subgroup of the whole diagram.
What we need is some way to measure the effect this has on the rest of the group.


Coxeter-Todd Algorithm
----------------------

This is exactly what the [*Coxeter-Todd algorithm*](
    https://en.wikipedia.org/wiki/Todd%E2%80%93Coxeter_algorithm
  ) does.
It more-or-less generalizes the procedure for creating Cayley graphs.
Using the vertices of the Coxeter diagram as generators[^2], we convert products between
  them to paths in a graph.

[^2]: For clarity, I'll try to refer to these vertices specifically as "generators".

Let's apply the algorithm to the diagram $A_3$.
First, we need to label our generators as *a*, *b*, and *c*.
Then, we select a select a generator to remove and create a graph according to the following steps:


### 1. Initialization

In a Cayley graph, we start out with a single vertex representing the identity.
Here, the first vertex instead corresponds to the subgroup generated by all other generators.

For each generator other than the one being removed, we attach loops to the vertex.
This is because the subgroup is closed -- when we multiply the set of all elements
  of the subgroup by any of its elements, it doesn't change the set.
In other words, $hH = H, h \in H$.

For the remaining generator, we draw an edge connecting to a new vertex,
  which represents a new set of elements.
This is a coset of the subgroup -- that is, each vertex we make in this new graph
  corresponds to a disjoint partition of elements of the whole group.

![
  Left: the Coxeter diagram $A_3$ <br>
  Right: the first step of generating a graph from removing generator *a*.
  The (group generated by the) *bc* subdiagram is represented by the vertex with the *b* and *c* loops.
](./steps/coxeter_todd_1.png)

Note that all loops and edges we draw will be undirected, since all generators have order 2.
Performing the same operation twice just takes us back to where we started.


### 2. Loop Transference

At the new vertex, we transfer over every loop whose label is not connected to the label of the edge.
This is due to the rule stating that non-adjacent generators commute.
In other words, their product has order 2.
For our graph, this can be seen for $acac = a^2 c^2 = \text{id} = (ac)^2$, and means after following
  labels *acac*, we end up at the same vertex we started at.

![&nbsp;](./steps/coxeter_todd_step_2.png)


### 3. Extension, Branching, and Rejoining

At the new vertex, we still need an edge labelled *b*, but it cannot be a loop.
Adjacent generators have a product whose order is the label (*n*) of the edge connecting them
  (defaulting to 3 for unlabelled ones).
This means we need to form a path alternating between the two generators,
  such that following it would loop back to the vertex we started at.
That is,

$$
(ab)^n = \stackrel{n \text{ times}}{\overbrace{(ab)(ab) ... (ab)(ab)}} = \text{id}
$$

![
  Attaching a *b* edge to a new vertex with an *a* loop so that $ababab = \text{id}$
](./steps/coxeter_todd_step_3.png)

If the Coxeter diagram branches, then so does the generated graph.
The labels of each pair of branching edges have a product with a known order,
  so following a path alternating between two of them must return to the same vertex.
Since it alternates between the two labels *n* times, this ends up producing even-sided polygons
  in our generated graph.
Commuting vertices form either a square or a pair of loops connected by an edge.
The latter is actually a special case of the former, which can be seen by matching
  a pair of opposite edges.

![
  Top: two nonadjacent vertices form a square <br>
  Bottom: two adjacent vertices form a hexagon
](./steps/coxeter_todd_cycles.png){.narrower}


### 4. Finalization

If we ensure at each new vertex that we can follow a valid path for every pair of generators,
  then this process will either terminate or continue indefinitely with a repeating pattern.
A quick sanity check on the resulting graph is to add the number of loops and edges at a vertex.
The sum should be the number of generators.

![
  Final graph to the middle right. The *ab* and *bc* paths are shown above and below.
](./steps/coxeter_todd_step_4.png){.narrower}

As stated earlier, each vertex in the graph corresponds to a coset of the group
  under the subgroup being considered.
The number of cosets is the index of the subgroup, the quantity we were after the whole time.
In this case, the index of the three-vertex diagram ($A_3$) on the two-vertex diagram
  ($A_2$, the same diagram without *a*) is 4.


### Results

We can remove vertices one-by-one until we eventually reduce the graph to a single vertex,
  which we know has order 2.
Multiplying all of the indices at each stage together gives us the order of the group we are after.

![
  Coxeter diagrams (left) and the graphs generated by removing the a vertex (right). <br>
  Permutations shown above vertices which give a permutation group embedding.
](./coxeter_an.png){.narrow}

It is readily shown that the group generated by $A_3$ has order $4 \cdot 3 \cdot 2 = 24$,
  which coincides with the order of $S_4$.

We can assign a permutation to each of our generators, so long as the edge condition
  as it relates to their product is obeyed.
In the diagrams above, $(1 ~ 2)$ and $(2 ~ 3)$ have a product of order 3,
  but $(1 ~ 2)$ and $(3 ~ 4)$ have a product of order 2,
  since the cycles (and in the diagram, the vertices) are disjoint.
This argument generalizes nicely to higher $A_n$.


Other Diagrams for $A_3$
------------------------

A path is hardly the most interesting diagram.
However, there are yet more graphs which can be produced from $A_3$.


### Starting in the Middle

Selecting either endpoint of the diagram (generator *a* or *c*) will generate the same graph as shown above.
This is easy to see if you read the generated graph right-to-left rather than left-to-right.
However, the center point *b* has two neighbors.
If we try generating a graph by removing this generator, our graph branches.

![&nbsp;](./coxeter_a3_generator_b.png){.narrow}

The index of 6 given by this graph is correct, since the remaining diagram
  $A_1 A_1$ has an order of $2 \cdot 2 = 4$.
This group is isomorphic to the [Klein four-group](https://en.wikipedia.org/wiki/Klein_four-group),
  so this diagram shows that it occurs within $S_4$.


### Removing Multiple Vertices

We can remove more than one generator from the Coxeter diagram at once, as long as we follow all the rules.
Instead, the initial vertex in the new graph branches immediately and has fewer loops.
For example, removing two generators from $A_3$ produces a figure with squares and hexagons:

![
  Graph generated by removing any two vertices from $A_3$. <br>
  For the circled generators *a* and *b*, the diagram starts from the circled vertex
    on the right (or its identical neighbor). <br>
  If *b* and *c* were removed instead, the diagram would start from
    the similar vertices with *a* loops at the top of the graph. <br>
  If *a* and *c* were removed, the diagram would start from either the leftmost or rightmost vertex,
    which both have a *b* loop.
](./coxeter_a3_pair.png){.narrow}

A figure composed of squares and hexagons should be familiar from the previous post.
The truncated octahedron also contains only squares and hexagons and is generated by $S_4$.
In fact, the figure above is the truncated octahedron folded in half.
It is folded in "half" specifically because the single remaining vertex in the Coxeter diagram has order 2.
Consequently, the number of vertices in the graph must be half that
  of the whole group (or complete permutahedron).

If we remove both generators in $A_2$, we end up with a hexagon.

![
  Graph generated by removing both generators from $A_2$.
  All vertices are identical, so all may be considered "starting" vertices.
](./coxeter_a2_both.png){.narrow}

Also from the previous post, we know this to be the Cayley graph for $S_3$ generated by a pair of swaps.
In this way, the Cayley graph can be thought of as a sort of "limiting" graph of the Coxeter diagram,
  where each coset contains only a single element.


The Traditional Explanation
---------------------------

I must now admit that that I have not given anywhere near the whole explanation of Coxeter diagrams,
  and have only succeeded in leading us down the proverbial algebraic rabbit hole.

Coxeter diagrams also come equipped with a standard geometric interpretation.
The "order 2 elements" we discussed previously are replaced with mirrors,
  and the edge labels are the angle between mirrors,
  interpreted as half of the central angle of a regular *n*-gon (triangle: 60°, square: 45°, ...).

<!-- TODO: better wikimedia imports-->
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
  <a title="Tomruen, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons" href="https://commons.wikimedia.org/wiki/File:Dihedral_symmetry_domains_3.png">
    <img width="192" alt="Dihedral symmetry domains 3" src="https://upload.wikimedia.org/wikipedia/commons/e/e2/Dihedral_symmetry_domains_3.png">
  </a>

  <figcaption>
  Fundamental domains of Coxeter diagram ![](coxeter_a2.png)
  <br>
  Source: Tomuren, via Wikimedia Commons
  </figcaption>
</figure>
</div>

The image above is composed of the mirrors specified by $A_2$.
One of the generators is the red reflection and the other is the the green reflection.
It is clear that alternating between red and green three times will take one back
  to where they started, just like the algebra tells us.

All of the plane ends up assigned to one of the domains.
Note also that points along mirrors which are equidistant from the center
  form a hexagon made up of six equilateral triangles, one in each domain.


### Struggles in 3D

$A_3$ has an additional vertex and edge.
This means that an additional mirror needs to be placed in along every preexisting reflection.
We end up reflecting into another dimension, and our domains shoot out into 3D space.
It is easier to imagine the additional domains by rotating two of the axes around each other,
  so that the pairs will join together into equilateral triangular "cones".

![
  Mirror axes in 3D for $A_3$.
  The red, green, and blue lines all lie within the same plane, and correspond to $A_2$.
  The cyan line is perpendicular to the red one, and is formed by rotating
    either green about blue or blue about green by 60°.
](./roots/roots_a3.png){.narrowest}

In 2D, equidistant points form a hexagon; in 3D, the points instead form a cuboctahedron.
Connecting the triangular faces of the cuboctahedron to the origin gives
  a family of eight tetrahedra meeting at a point.
At this point the reflections become clearer, since certain reflections
  do the same thing as in the plane above, while others will "double" a tetrahedron.

::: {layout-ncol="2" layout-valign="center"}
![Cuboctahedron](./roots/cuboctahedron.png)

![
  Six of the eight tetrahedra formed by connecting equidistant points ("roots") to the origin.
  Note how whole tetrahedra are reflected in a mirror axis.
](./roots/roots_a3_with_domains.png)
:::

Personally, I think the geometric way of thinking becomes is difficult even in 3D.
Every axis describes a separate rotation, which is simple enough.
But we're trying to create infinite cones of symmetry domains in 3D space,
  and we can only try to interact with this using a 2D screen.
We're running out of room before even getting to 4D space,
  where the visualization problems are even worse.
Also, while the angle between two mirrors is 60°, it's easy to slip into thinking
  about the dihedral angle between two faces (domains) of the cones, which is *not* 60°.


Closing
-------

When I initially wrote the article about swap diagrams (a name I hadn't given them at the time)
  I had no idea about how closely it related to something else which I was having trouble understanding.
Now knowing better, I split the article and pushed the basics a little harder.
I find this closeness between swap diagrams and Coxeter diagrams interesting.
At the cost of simplicity, the latter enables the ability to encode higher-order cycles and give rise
  to exceptional mathematical objects.

It was difficult for me to find information documenting how to understand the algebraic perspective.
It also bears mentioning that I know very little about Lie theory,
  another field of math where Coxeter diagrams turn up.
When I finally found [a group theory video](https://www.youtube.com/watch?v=BHezLvEH1DU)
  by Professor Richard Borcherds, I was dismayed that there seems to be even less material
  online which depicts the coset graphs.
I have written [an appendix](../appendix) cataloguing some graphs that appear when applying
  the Coxeter-Todd algorithm, since I believe they are interesting in their own right.

Images from Wikimedia belong to their respective owners.
All other diagrams created with GeoGebra.
