Type D root system - Knowlpedia

Definition

For $n\geq4$ , the root system of type $D_n$ is

D_n=\{\pm e_i\pm e_j:1\leq i<j\leq n\}\subset\mathbb R^n,

where the two signs are chosen independently. It is an irreducible simply laced root system of rank $n$ , with $2n(n-1)$ roots.

Simple roots and diagram

One standard base is

\alpha_i=e_i-e_{i+1}\quad(1\leq i<n), \qquad \alpha_n=e_{n-1}+e_n.

The Dynkin diagram is a chain that forks into two terminal vertices at one end. Its Weyl group consists of permutations of the coordinates together with sign changes of an even number of coordinates; it has order $2^{n-1}n!$ .

Lie-algebra realization

Type $D_n$ is the root system of the complex orthogonal Lie algebra

\mathfrak{so}_{2n}(\mathbb C),

which has dimension $n(2n-1)$ . Thus $D_n$ is the even-orthogonal family in the classification of complex simple Lie algebras.

Low-rank coincidences

Extending the coordinate formula to small $n$ gives reducible or coincident systems:

D_2\cong A_1\sqcup A_1, \qquad D_3\cong A_3.

The first genuinely new irreducible case is $D_4$ , whose diagram has an order-six symmetry responsible for triality.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§11–12. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §4. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §§5–6. arXiv record.