How to read Dynkin diagrams

Dynkin diagrams are a compact method of encoding Cartan matrices, but it can be challenging to try to remember which way around the arrows go. This is not usually an issue when working theoretically, but in examples or programming where everything has to be explicit, getting things around the right way is important.

5 kinds of junctions

The convention used here is that the Cartan matrix A = [a_{ij}] associates rows with coroots and columns with roots, so we have a_{ij} = \innprod{\alpha_i^\vee, \alpha_j}. Equivalently, in some Weyl-invariant symmetric bilinear form (-, -) we have a_{ij} = 2 \frac{(\alpha_i, \alpha_j)}{(\alpha_i, \alpha_i)}. As far as I am aware, this convention is standard throughout the Kac-Moody and quantum groups literature, though sometimes differs from the conventions used in the literature on Coxeter groups.

In the table below, we consider to have fixed the ordering i < j, so that i is the left vertex in the Dynkin diagram, and [a_{ij}] = \begin{pmatrix} a_{ii} & a_{ij} \\ a_{ji} & a_{jj} \end{pmatrix} = \begin{pmatrix} \innprod{\alpha_i^\vee, \alpha_i} & \innprod{\alpha_i^\vee, \alpha_j} \\ \innprod{\alpha_j^\vee, \alpha_i} & \innprod{\alpha_j^\vee, \alpha_j} \end{pmatrix}.

The following table lists the 5 kinds of junctions (6, if we include A_1 \times A_1 as a junction) which are found in the finite and affine type Dynkin diagrams.

  • The column a_{ij} a_{ji} records the product of the off-diagonal entries, which determines m_{ij}.
  • The number m_{ij} is the order of (s_i s_j) in the Weyl group, so that (s_i s_j)^{m_{ij}} = 1.
  • The matrix [i \cdot j] is the smallest integral symmetrisation of [a_{ij}] such that the diagonals are positive even integers. The diagonal entries record the square lengths of roots. We can see that the arrow always points to the smaller root, and that the number of lines is the ratio of the square lengths of the roots.

There are three kinds of nontrivial junction with m_{ij} < \infty, which come from the A_2, B_2 or C_2, and G_2 Dynkin diagrams. The G_2 junction appears exclusively in finite type G_2, and affine types G_2^{(1)} and and D_4^{(3)}. The other two kinds of junction seen in affine type are the A_1^{(1)} kind and A_2^{(2)} kinds, seen only in those exact Cartan types.

In Lusztig’s Introduction to quantum groups, the matrix [i \cdot j] is called a Cartan datum, and is used in place of the Cartan matrix. This has the effect of forcing all Cartan matrices to be symmetrisable, and is further important because the choice of normalisation i \cdot i features in the definition of the quantum group.

Conventions used in books

The books I know of that use the same a_{ij} = \innprod{\alpha_i^\vee, \alpha_j} convention on this page are:

  1. Infinite-dimensional Lie algebras by Victor Kac, 3rd Edition (1990). a_{ij} introduced in Section 1.1.
  2. Quantum groups and their primitive ideals by Anthony Joseph (1995). In section 3.1 we have a_{ij} = (\alpha_i^\vee, \alpha_j).
  3. Lectures on Quantum Groups by Jens Carsten Jantzen (1996). At the start of Chapter 4, we have a_{\alpha \beta} = 2 (\alpha, \beta) / (\alpha, \alpha).
  4. Kac-Moody groups, their flag varieties and representation theory by Shrawan Kumar (2002). In definition 1.1.2 we have a_{ij} = \alpha_j(\alpha_i^\vee).
  5. Introduction to quantum groups and crystal bases by Hong and Kang (2002). In section 2.1 we have a_{ij} = \alpha_j(h_i).
  6. Kashiwara’s papers on crystals and quantized enveloping algebras.

A very notable exception which uses the opposite convention is Bourbaki, who sets n(\alpha, \beta) = 2 (\alpha | \beta) / (\beta | \beta) (VI, Section 1, part 5, definition 3).