A Weyl Group is a group associated with a compact Lie group that can either be abstractly defined in terms of a root system or in terms of a maximal torus. More generally, there are Weyl groups associated with symmetric spaces. They can also be viewed as a special type of finite Coxeter group, i.e. a group generated by reflections which, in the case of Weyl groups, acts discretely by isometries on a sphere in some dimension.
The Weyl group of symmetries of a root system. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
Definition: Given a compact Lie group $~G~$ with chosen maximal torus $~T~$, its Weyl group $~W(G)=W(G,T)~$ is the group of automorphisms of $~T~$ which are restrictions of inner automorphisms of $~G~$.
This is the quotient group of the normalizer subgroup of $~T⊂G~$ by $~T~$ $$W≃N_G(T)/T~.$$
Properties:
- The maximal torus is of finite index in its normalizer; the quotient $~N(T)/T~$ is isomorphic to $~W(G)~$.
- The cardinality of $~W(G)~$ for a compact connected $~G~$, equals the Euler characteristic of the homogeneous space $~G/T~$.
- An important approach to the representations of the Weyl groups is the Springer theory.
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