Weylryhmä

Weylryhmä is a mathematical concept in the field of Lie theory. It is a finite group associated with a given root system, which in turn is related to a Lie algebra or a Lie group. The Weyl group acts as a symmetry group on the root system. Specifically, it is generated by reflections across hyperplanes that are orthogonal to the roots. These reflections permute the roots in a way that preserves the root system's structure. The Weyl group plays a crucial role in understanding the structure and representations of Lie algebras and Lie groups. It helps classify simple Lie algebras and provides a framework for studying their irreducible representations. For instance, the Weyl group is instrumental in determining the dimensions of irreducible representations and in formulating the Weyl character formula. The concept was introduced by Hermann Weyl. Different types of root systems, such as A, B, C, D, E, F, G, correspond to distinct Weyl groups. The order of the Weyl group is related to the number of roots and their interrelations. The Weyl group is a fundamental tool in various areas of mathematics and physics, including algebraic geometry, representation theory, and quantum field theory.