Weyl Group

A Weyl group is a finite group of symmetries associated with a root system, generated by the reflections that fix a hyperplane through the origin for each root, and it captures the essential discrete symmetry underlying a semisimple Lie algebra or Lie group. The groups are named after the German mathematician Hermann Weyl, who developed much of the representation theory of Lie groups in the 1920s, building on earlier structural work by Wilhelm Killing and Elie Cartan on the classification of simple Lie algebras. Every Weyl group is an example of a finite Coxeter group, a group generated by reflections satisfying simple relations, and the Weyl group of the root system of type A(n) is isomorphic to the symmetric group on n plus one letters, linking the abstract theory directly to ordinary permutations. Weyl groups play a central role in the classification of semisimple Lie algebras and in the representation theory built on top of that classification.