Topic Cartan Matrices and Dynkin Diagrams
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Cartan Matrices and Dynkin Diagrams

The Cartan matrix and Dynkin diagram are combinatorial distillations of a root system: the matrix encodes pairwise root relations as integers, and the diagram is its pictorial encoding. Together they classify finite-type root systems and, with them, the simple algebraic groups.

Cartan matrix

Let \(\Phi\) be a root system with base \(\Delta = \{\alpha_1, \ldots, \alpha_r\}\). The Cartan matrix is \[ C \;=\; (c_{ij})_{1 \le i, j \le r}, \qquad c_{ij} \;=\; \langle \alpha_i, \alpha_j^\vee \rangle. \]

Equivalently: \(c_{ij}\) is the integer such that the reflection \(s_j\) acts on \(\alpha_i\) by \(s_j(\alpha_i) = \alpha_i - c_{ij} \alpha_j\).

Properties:

  • \(c_{ii} = 2\) for all \(i\);
  • \(c_{ij} \in \{0, -1, -2, -3\}\) for \(i \ne j\);
  • \(c_{ij} = 0 \iff c_{ji} = 0\) (orthogonality is symmetric);
  • The matrix \((c_{ij} + c_{ji})\) is positive-definite (the positive-definite property of \(\Phi\)).

Dynkin diagram

The Dynkin diagram is a finite graph encoding the Cartan matrix:

  • One vertex per simple root \(\alpha_i\).
  • Between distinct \(\alpha_i, \alpha_j\): edges encoded by \(c_{ij} \cdot c_{ji} \in \{0, 1, 2, 3\}\):
  • 0: no edge,
  • 1: single edge,
  • 2: double edge,
  • 3: triple edge.
  • Arrow on multi-edges: pointing from the longer root to the shorter (in standard normalisation \((\alpha_i, \alpha_i) \ge (\alpha_j, \alpha_j)\)).

A Dynkin diagram is connected iff the root system is irreducible (equivalently, the corresponding Lie algebra/simple group is simple).

Classification

Theorem (Killing, Cartan, Dynkin). The connected Dynkin diagrams of finite-type irreducible root systems are exactly:

Type Diagram Rank
\(A_n\) line of \(n\) vertices, all single \(n \ge 1\)
\(B_n\) line, double edge at one end pointing in \(n \ge 2\)
\(C_n\) line, double edge at one end pointing out \(n \ge 3\)
\(D_n\) "Y-shape": line with a forked end \(n \ge 4\)
\(E_6, E_7, E_8\) "T-shape": three legs of varying lengths 6, 7, 8
\(F_4\) line of 4 with double edge in middle 4
\(G_2\) two vertices, triple edge 2

Examples

Type \(A_2\) (corresponding to \(\mathrm{SL}_3\))

Cartan matrix: \[ C \;=\; \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}. \] Diagram: two vertices joined by a single edge \(\bullet - \bullet\).

Type \(B_2 = C_2\) (\(\mathrm{Sp}_4 = \mathrm{Spin}_5\))

Cartan matrices: \[ B_2: \begin{pmatrix} 2 & -1 \\ -2 & 2 \end{pmatrix}, \qquad C_2: \begin{pmatrix} 2 & -2 \\ -1 & 2 \end{pmatrix}. \] Diagram: two vertices, double edge \(\bullet = \bullet\) with arrow direction encoding short/long roots.

Type \(G_2\)

Cartan matrix: \[ C \;=\; \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}. \] Diagram: two vertices, triple edge \(\bullet \equiv \bullet\).

Type \(E_8\)

Cartan matrix is \(8 \times 8\); diagram is a "T"-shape: \(\bullet - \bullet - \bullet - \bullet - \bullet - \bullet - \bullet\) with one extra vertex attached to the third from one end.

Duality on diagrams

The dual diagram swaps:

  • \(B_n \leftrightarrow C_n\) (arrow reversal in the diagram);
  • \(F_4\) is self-dual under arrow reversal (sends \(F_4\) to itself via the central symmetry).

All other types are self-dual under root-system duality (the arrow reversal is trivial since there are no double edges, or coincides with the type).

The Langlands dual group corresponds to this swap on the dual side.

Connection to Coxeter diagrams

A Coxeter diagram records only the orders of the products of simple reflections: \[ m_{ij} \;=\; \mathrm{ord}(s_i s_j) \in \{2, 3, 4, 6\}. \]

The Dynkin diagram has more information than the Coxeter diagram: it distinguishes \(B_n\) from \(C_n\) (same Coxeter, different Dynkin).

Reflection groups and finite-type Coxeter groups

The Coxeter group of a Dynkin diagram is a finite reflection group, classified by:

\[ A_n, \quad B_n = C_n \text{ (same Coxeter)}, \quad D_n, \quad E_6, E_7, E_8, F_4, G_2, \]

plus non-crystallographic types \(H_3, H_4, I_2(p)\) that are not realised by root systems with integral Cartan matrix.

Mathlib

Mathlib.LinearAlgebra.RootSystem.IsCrystallographic and related modules formalise the Dynkin-classification side; the diagram classification (which combinations of A, B, C, D, E, F, G can occur) is partly there.

References

  • Bourbaki, Groupes et Algèbres de Lie IV–VI (1968).
  • Humphreys, Introduction to Lie Algebras and Representation Theory (1972), §3, §11.
  • Carter, Finite Groups of Lie Type (1985), §1.
  • Serre, Complex Semisimple Lie Algebras (1987).

Prerequisite Topics

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