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## Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of Humphreys.

A Coxeter system $(W, S)$ is a group $W$ together with a generating set $S$ and presentation of the form

$\langle s_1, ... s_n | (s_i s_j)^{m(i, j)} = 1 \rangle$

where $m(i, j) = m(j, i), m(i, i) = 1$, and $m(i, j) \ge 2, i \neq j$. (When there is no relation between $s_i, s_j$, we write this as $m(i, j) = \infty$.) The group $W$ is a Coxeter group, and is usually understood to come with a preferred presentation, so we will often abuse terminology and use “group” and “system” interchangeably. $S$ is also referred to as the set of simple reflections in $W$, and $n$ the rank. (We will only consider finitely-generated Coxeter groups.)

Historically, Coxeter groups arose as symmetry groups of regular polytopes and as Weyl groups associated to root systems, which in turn are associated to Lie groups, Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a non-technical overview of Coxeter groups and $q$-analogues, I recommend John Baez’s week184 through week187. The slogan you should remember is that Weyl groups are “simple algebraic groups over $\mathbb{F}_1$.”

## The McKay correspondence I

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 $\text{SU}(2)$, 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 $G$ be a finite subgroup of $\text{SL}_2(\mathbb{C})$. Since any inner product on $\mathbb{C}^2$ can be averaged to a $G$-invariant inner product, every finite subgroup of $\text{SL}_2(\mathbb{C})$ is conjugate to a finite subgroup of $\text{SU}(2)$, so we’ll suppose this without loss of generality. The two-dimensional representation $V$ of $G$ coming from this description is therefore faithful and self-dual. Consider the representation graph $\Gamma(V)$, whose vertices are the irreducible representations of $G$ and where the number of edges between $V_i$ and $V_j$ is the multiplicity of $V_j$ in $V_i \otimes V$. We will see that $\Gamma(V)$ is a connected undirected loopless graph whose spectral radius $\lambda(\Gamma(V))$ is $2$. Today our goal is to prove the following.

Theorem: The connected undirected loopless graphs of spectral radius $2$ are precisely the affine Dynkin diagrams $\tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$.

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.

## Dynkin diagrams and the Mahler measure problem

Funnily enough, a few days after I wrote the previous post, I was linked to a graph theory paper where one of the first results cited, which was clearly well-known to the authors, is the following remarkable generalization of what I tried to do:

Theorem: The only connected simple graphs with spectral radius less than or equal to $\ 2$ are the induced subgraphs of the Dynkin diagrams $\tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$.

I have to admit, I really didn’t suspect that the classification result I was going after was both so simple and so interesting! Certainly there are heuristic reasons why the above classification makes sense: as I forgot to note in the previous post, there really can’t be too many vertices of degree $3$ in a graph with $\rho(G) \le 2$. But I really can’t fathom why spectral radius can be used to define the Dynkin diagrams, considering their relationship to

binary polyhedral groups and the Platonic solids,

the octonions (okay, this one is stretching it a little).

Anyone know any good references?

In any case, I’d like to discuss the McKee-Smyth paper because it has some interesting ideas I’d thought about independently.