Classification of Irreducible Root Systems

The Killing–Cartan classification theorem, completed by Dynkin, gives a complete list of finite irreducible reduced crystallographic root systems. These correspond bijectively to the connected Dynkin diagrams of finite type.

The classification theorem

Theorem (Killing 1888; Cartan 1894; Dynkin 1947). Every finite irreducible reduced crystallographic root system is isomorphic to exactly one of the following:

Classical types

• \(A_n\) (\(n \ge 1\)): Dynkin diagram is a linear chain of \(n\) nodes connected by single bonds: \(\circ - \circ - \cdots - \circ\) (\(n\) nodes). Root system in \(\mathbb{R}^n \cap \{\sum x_i = 0\}\): \(\Phi = \{e_i - e_j : i \ne j\}\), with \(n(n+1)\) roots. Weyl group: \(S_{n+1}\), order \((n+1)!\).

• \(B_n\) (\(n \ge 2\)): Linear chain of \(n-1\) single bonds followed by one double bond with arrow pointing right (short root): \(\circ - \circ - \cdots - \circ \Rightarrow \circ\). \(\Phi = \{\pm e_i \pm e_j : i < j\} \cup \{\pm e_i\}\), with \(2n^2\) roots. Long roots: \(\pm e_i \pm e_j\); short roots: \(\pm e_i\).

• \(C_n\) (\(n \ge 3\)): Linear chain with one double bond at the left, arrow pointing left (long root): \(\circ - \circ - \cdots - \circ \Leftarrow \circ\) (or equivalently \(\circ \Rightarrow \circ - \cdots - \circ\) depending on convention). \(\Phi = \{\pm e_i \pm e_j : i < j\} \cup \{\pm 2e_i\}\), with \(2n^2\) roots. Long roots: \(\pm 2e_i\); short roots: \(\pm e_i \pm e_j\).

• \(D_n\) (\(n \ge 4\)): Linear chain of \(n-2\) nodes, then forking into two branches at the end: \(\circ - \circ - \cdots - \circ < \!\!\! ^\circ_\circ\). \(\Phi = \{\pm e_i \pm e_j : i < j\}\), with \(2n(n-1)\) roots. All roots have equal length; simply laced.

Exceptional types

• \(G_2\): Two nodes connected by a triple bond with arrow: \(\circ \!\Rrightarrow\! \circ\). 12 roots; two root lengths in ratio \(\sqrt{3}:1\). Weyl group dihedral of order 12. Unique among rank-2 crystallographic root systems.

• \(F_4\): Four nodes: \(\circ - \circ \Rightarrow \circ - \circ\). 48 roots; two root lengths in ratio \(\sqrt{2}:1\). Weyl group of order 1152.

• \(E_6\): A chain of 5 nodes with one branch off the third: \(\circ - \circ - \circ - \circ - \circ\) with \(\circ\) above the middle. 72 roots; simply laced. Weyl group of order 51840.

• \(E_7\): A chain of 6 nodes with one branch off the third from left. 126 roots; simply laced. Weyl group of order 2903040.

• \(E_8\): A chain of 7 nodes with one branch off the third from left. 240 roots; simply laced. Weyl group of order 696729600.

Root counts

Note: \(B_2 \cong C_2\) and \(A_3 \cong D_3\) as root systems; the constraints \(n \ge 2\) for \(B\), \(n \ge 3\) for \(C\), \(n \ge 4\) for \(D\) eliminate these overlaps.