Properties of The Irreducible Root Systems
| I | D | ||||
|---|---|---|---|---|---|
| An (n ≥ 1) | n(n + 1) | n + 1 | (n + 1)! | ||
| Bn (n ≥ 2) | 2n2 | 2n | 2 | 2 | 2n n! |
| Cn (n ≥ 3) | 2n2 | 2n(n − 1) | 2 | 2 | 2n n! |
| Dn (n ≥ 4) | 2n(n − 1) | 4 | 2n − 1 n! | ||
| E6 | 72 | 3 | 51840 | ||
| E7 | 126 | 2 | 2903040 | ||
| E8 | 240 | 1 | 696729600 | ||
| F4 | 48 | 24 | 4 | 1 | 1152 |
| G2 | 12 | 6 | 3 | 1 | 12 |
Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (An, Bn, Cn, and Dn, called the classical root systems) and five exceptional cases (the exceptional root systems). The subscript indicates the rank of the root system.
In an irreducible root system there can be at most two values for the length (α, α)1/2, corresponding to short and long roots. If all roots have the same length they are taken to be long by definition and the root system is said to be simply laced; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to r2/2 times the coroot lattice, where r is the length of a long root.
In the table to the right, |Φ < | denotes the number of short roots, I denotes the index in the root lattice of the sublattice generated by long roots, D denotes the determinant of the Cartan matrix, and |W| denotes the order of the Weyl group.
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