irreps

In physics and mathematics, "irreps" is a common shorthand for irreducible representations. Irreducible representations are fundamental building blocks in the study of group theory, particularly in areas like quantum mechanics, particle physics, and crystallography. A representation of a group is a way to map the elements of the group to invertible matrices. These matrices then act on a vector space. A representation is called "irreducible" if the only subspaces of the vector space that are invariant under the action of all the group's matrices are the trivial subspace (containing only the zero vector) and the entire vector space itself. In essence, an irreducible representation cannot be broken down into smaller, independent representations. This property makes them crucial for analyzing the symmetries of physical systems. For instance, in quantum mechanics, the irreducible representations of the rotation group describe the different possible angular momenta of particles. Similarly, in particle physics, the irreducible representations of symmetry groups classify fundamental particles and their properties. The classification and understanding of irreducible representations for various groups are central to many theoretical frameworks.