Order from Coxeter-Dynkin diagram

Question

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?

Answer 1

First of all, you may as well assume your diagram is connected, otherwise the order will be the product of the orders of the irreducible components. I'll assume this from now on.

As Nathan Reading mentioned, it is well-known that the order of the Coxeter group $W$ is $$ \#W = d_1d_2\cdots d_r$$ where $d_1,\ldots,d_r$ are the degrees of $W$ (see e.g. Humphreys's "Reflection groups and Coxeter groups," Section 3.9). Note we have $d_i = e_i+1$, where $e_1,\ldots,e_r$ are the exponents of $W$. But it is not so clear how to immediately read off the degrees/exponents from the Dynkin diagram.

In the case where $W$ is a Weyl group, there is a different formula for the order of $W$: $$ \#W = f \cdot r! \cdot a_1\cdots a_r.$$ Here $r$ is the rank of $W$, $f$ is the index of connection, and the $a_i$ are the coefficients expressing the highest root $\theta$ as a sum of simple roots: $\theta = a_1\alpha_1 + \cdots + a_r\alpha_r$. See Section 4.9 of the aforementioned book of Humphreys, or Lam-Postnikov "Alcoved Polytopes II" for a $q$-analog.

So how can we read this information from the Dynkin diagram? Of course $r$ is just the number of nodes; and $f$ is the determinant of the Coxeter matrix (it is also equal to one plus the number of $a_i$ equal to one). But what about the $a_i$?

To find the $a_i$ we can iteratively compute $\theta$ as follows. We start by setting $\alpha=\alpha_i$ for some simple root $\alpha_i$. As long as the height of $s_{j}(\alpha)$ for some simple reflection $s_j$ is greater than the height of $\alpha$, replace $\alpha$ by $s_{j}(\alpha)$. Note that this can all be easily phrased algorithmically in terms of vectors in $\mathbb{Z}^r$ and rows of the Coxeter matrix, so indeed is easy to implement just from the Dynkin diagram.

Eventually we terminate at some $\alpha$ for which every simple reflection decreases our height. If the Dynkin diagram is simply laced, this $\alpha$ must be the highest root $\theta$ and then we've found the $a_i$. In the non-simply laced case, we either terminate at the highest root (if we started with a long simple root) or the highest short root (if we started at a short simple root). So we either need to make sure we start at a long simple root, or just try starting at every root to find the highest root we can.

Note that there is another description of the $a_i$ in terms of additive functions on the extended Dynkin diagram (+ subadditive functions on the Dynkin diagram). Basically, $(a_0,a_1,\ldots,a_r)$ is the unique primitive additive function on the extended Dynkin diagram (where always $a_0=1$).

Answer 2

There is, as far as I know, no way to compute this, in one step, from the diagram. But there is a recursive formula that uses the diagram and prior knowledge of the length of the longest element. (Acknowledgment: I learned about this from John Stembridge and looked it up in some notes he shared with me years ago.) The recursion runs over all subsets of the nodes of the diagram, so it's not efficient, but for specific infinite families, it can run smoothly. We need to give Poincaré series rather than just orders.

For a Coxeter system $(W,S)$, let $W(q)=\sum_{w\in W}q^{\ell(w)}$, where $\ell$ is the usual length function relative to words in $S$. For $J\subseteq S$, let $W_J$ be the standard parabolic subgroup generated by $J$, so that $W_J(w)=\sum_{w\in W_J}q^{\ell(w)}$.

Theorem (Steinberg).
$$\sum_{J\subseteq S}(-1)^J\frac{1}{W_J(q)}=\begin{cases}\frac{q^{\ell(w_0)}}{W(q)}\text{ if }W\text{is finite with longest element }w_0,\\0\text{ if }W\text{ is infinite.}\end{cases}$$

This theorem is supposed to be in Steinberg's monograph Endomorphisms of linear algebraic groups (AMS Memoir 80), but I haven't chased that down.

It's useful to remember that if $W$ has a disconnected diagram, the its Poincaré series is the product of the Poincaré series of its components.

Answer 3

Several answers to this question are given in Section 12 of a paper called "Generalized cluster complexes and Coxeter combinatorics" (https://arxiv.org/abs/math/0505085). Specifically, there are four (more or less crazy) ways to determine invariants including exponents recursively from the diagram (if you know all invariants for connected subgraphs of the diagram). Once you know the exponents, you know the order of $W$.

These methods are much too involved to describe here, but several of them have to do with the Fuss-Catalan number associated to $W$. This is a polynomial in a parameter $m$ and has a product formula in terms of the Coxeter number and exponents. You can factor the Fuss-Catalan number (polynomial) completely and (if you know the Coxeter number $h$) read off the exponents.