Today we’re going to relate the representation graphs introduced in this blog post to something I blogged about in the very first and second posts in this blog! The result will be a beautiful connection between the finite subgroups of , the Platonic solids, and the ADE Dynkin diagrams. This connection has been written about in several other places on the internet, for example here, but I don’t know that any of those places have actually gone through the proof of the big theorem below, which I’d like to (as much for myself as for anyone else who is reading this).

Let be a finite subgroup of . Since any inner product on can be averaged to a -invariant inner product, every finite subgroup of is conjugate to a finite subgroup of , so we’ll suppose this without loss of generality. The two-dimensional representation of coming from this description is therefore faithful and self-dual. Consider the representation graph , whose vertices are the irreducible representations of and where the number of edges between and is the multiplicity of in . We will see that is a connected undirected loopless graph whose spectral radius is . Today our goal is to prove the following.

**Theorem:** The connected undirected loopless graphs of spectral radius are precisely the affine Dynkin diagrams .

We’ll describe these graphs later; for now, just keep in mind that they are graphs with a number of vertices which is one greater than their subscript. In a later post we’ll see how these give us a classification of the Platonic solids, and we’ll also discuss other connections.