i, j

are joined by

a_{ij} a_{ji}

bonds, where

a_{ij}

are the off-diagonal Cartan integers. No bond means orthogonality; one, two and three bonds mean angles

2 π /3, 3 π /4, 5 π /6

. A multiple-bond arrow points to the shorter root. Thus the Dynkin diagram determines the Cartan matrix, including the relative lengths inside each connected component.

An irreducible root system corresponds to a connected Dynkin diagram. The classification of finite crystallographic Dynkin diagrams gives

A_{l} (l \geq 1), B_{l} (l \geq 2), C_{l} (l \geq 3), D_{l} (l \geq 4), E_{6}, E_{7}, E_{8}, F_{4}, G_{2} .

(1)

Here

C_{2}

is already represented by

B_{2}

, and

B_{1}, C_{1}

A_{1}

. General systems are orthogonal direct sums of these, with separate component scales. The following diagrams specify all the types; dots denote a continued chain, not additional vertices.

Figure 1.
Finite irreducible crystallographic Dynkin diagrams, with every arrow pointing toward the shorter root
. **Details**.

For orientation, the An Dynkin diagram is a single chain, the Bn Dynkin diagram has a terminal double bond pointing to its terminal short root, and the Cn Dynkin diagram reverses that arrow. The Dn Dynkin diagram has arm lengths

1, 1, l - 3

. The En Dynkin diagram has arm lengths

1, 2, 2

;

1, 2, 3

; or

1, 2, 4

. The F4 Dynkin diagram has a central double bond and the G2 Dynkin diagram a triple bond.

The positivity behind this finite list is the positive-definite Gram matrix of the simple roots. In particular the underlying graph is a tree: if it contained a cycle, the sum of its normalized roots would have squared norm at most zero, since each bonded inner product is at most

- 1/2

. This contradicts linear independence. Restricting the possible bonds and branches by positive definiteness gives precisely the listed chains and trees. Conversely their Cartan matrices are positive definite after symmetrization and their reflection-generated root sets give the indicated finite root systems.