Reflection groups are mathematical structures that consist of a set of reflections across hyperplanes in a vector space. These reflections can be combined to generate symmetries, which are important in various fields such as geometry and algebra. Reflection groups help in understanding the symmetrical properties of shapes and can be classified into finite and infinite groups. One of the most well-known types of reflection groups is the Coxeter groups, which are defined by specific rules regarding how reflections interact. These groups play a crucial role in the study of Lie algebras and root systems, providing insights into the underlying symmetries of mathematical objects.