Weylgroep

The Weyl group, or Weylgroep in Dutch, is a finite group associated with a root system in Lie theory. It consists of the symmetries of a semisimple Lie algebra that preserve its root system and weight lattice, acting by isometries on the ambient Euclidean space.

Formally, given a root system Φ in a Euclidean space V, for each simple root α_i there is a reflection s_i across the hyperplane orthogonal to α_i. The Weyl group W is the group generated by these reflections. W is a finite Coxeter group and encodes the symmetries of the Dynkin diagram and the arrangement of Weyl chambers, the connected components formed by the hyperplanes orthogonal to the roots.

Weyl groups are classified by Dynkin types: A_n, B_n, C_n, D_n, and the exceptional types E6, E7, E8, F4, and G2. The types B_n and C_n share the same Weyl group up to isomorphism, and D_n is a subgroup of index two in the B_n/C_n symmetry group for n ≥ 4. The orders of notable Weyl groups are: |W(A_n)| = (n+1)!, |W(B_n)| = |W(C_n)| = 2^n n!, |W(D_n)| = 2^{n-1} n!, and the exceptional types have fixed values such as |W(G2)| = 12, |W(F4)| = 1152, |W(E6)| = 51840, |W(E7)| = 2903040, and |W(E8)| = 696729600.

Applications of Weyl groups appear in representation theory (Weyl character formula), invariant theory, and geometry, where they describe symmetries of weight and root lattices and influence the combinatorics of representations and Schubert calculus.

See also: root system, Dynkin diagram, Weyl chamber, Coxeter group.