I'm reading "From Spinors to Quantum Mechanics" by Gerrit Coddens. In Chapter 2.9 he shows diagrams representing two different groups as nodes and edges making up a polyhedron in 3D space.
First he shows the Icosahedral group $Y$ as being represented by a truncated icosahedron.
Then he shows the symmetric group $S_4$ represented as a truncated cuboctahedron.
Unfortunately I wasn't able to follow his description of how these pictures were generated. My best understanding is something like this. The rotations and reflections that are symmetries of the icosahedron are the elements of the icosahedral group $Y$. These can be thought of as operators that act on points in 3D space. The idea is something about picking a point in 3D space to represent the identity. Then applying the group elements of $Y$ to this point and marking those as new points in the 3D space. This makes some sense to me. But I don't understand which nodes in the new picture get edges between them.
For $S_4$ I'm totally lost. We can pick a point in 3D space to represent the identity element of the group. But elements of $S_4$ don't act on vectors in 3D space, so I don't know how to get new points from old points.
- Can someone explain what is going on in these examples to me?
- Or could someone give me keywords to search so I could research more about this type of graphical representation of groups?
- Can any group be represented in 3D space in this way?
As pointed out in the comments, the graphs in question in the text are indeed Cayley graphs of the corresponding groups. A Cayley graph is generated by (1) identifying each element of the group with a vertex (2) identifying a generating set for the group and (3) for each element $g$ and each element $s$ of the generating set, include an edge between the vertices representing $g$ and $gs$.
There's probably exact details I'm missing here about directed edges, colors, etc., but these details don't seem important for the "From Spinors to Quantum Mechanics" section so I haven't gone into that level of detail yet.
Not all groups can be represented nicely in 3D this way. i.e. as simple polyhedra with non-intersecting faces and edges.