Cartan matrix ( $A_{ij}$ )

For ordered simple roots, the Cartan matrix has entries

A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩

. It determines their angles and relative lengths and therefore determines the corresponding crystallographic root system.

Table of contents
- Positive roots for a chain Cartan matrix Cartan matrix

Positive roots for a chain Cartan matrix ( $Φ^{+} = {α_{i} + \dots + α_{j} : 1 \leq i \leq j \leq l}$ )

 0  0

Suppose the Cartan matrix has diagonal two, entries minus one between adjacent indices, and all other entries zero. Normalize the simple-root squared lengths to two. Every root is a Weyl group translate of a simple root, so all roots have squared length two. If a positive root has coefficients

c_{1}, \dots, c_{l}

in the simple-root basis, let

c_{0} = c_{l + 1} = 0

. Then

2 = \sum_{i = 0}^{l} (c_{i + 1} - c_{i})^{2} .

(1)

The nonnegative integer coefficients force exactly one upward jump of one and one downward jump of one. Thus the coefficients are one on a consecutive interval and zero elsewhere. Conversely, an interval root can be extended by the next simple root: its Cartan pairing with that root is minus one, so the corresponding simple reflection adds it. Starting from a simple root constructs every interval and proves the displayed description. This also identifies the root system as type