Group representations

Group representations are mathematical tools used to understand symmetries by describing how a group—an abstract set of operations like rotations or reflections—can be applied to objects or spaces. Instead of working directly with the abstract elements, representations assign each symmetry operation a concrete linear transformation (like a matrix) acting on a vector space. This way, the complex symmetry behavior is translated into well-understood linear algebra, making it easier to analyze, classify, and utilize symmetries in physics, chemistry, and other sciences. In essence, group representations bridge abstract symmetry concepts and tangible mathematical actions.