Irreducible representations are mathematical objects used to describe the symmetry properties of a physical system. They are commonly used in the field of group theory and play a crucial role in understanding the behavior of quantum mechanical systems.
Irreducible representations provide a convenient way to classify and describe the symmetries of a physical system. They are essential in understanding the energy levels, selection rules, and other properties of molecules, crystals, and other quantum systems.
Irreducible representations are closely related to group theory, which is the mathematical study of symmetry. In group theory, a group is a set of elements that can be combined (or operated) in a certain way to produce other elements of the group. Irreducible representations are used to describe how these operations transform the elements of the group.
Reducible representations can be broken down into a combination of two or more irreducible representations, while irreducible representations cannot be further simplified. In other words, reducible representations contain multiple independent subrepresentations, while irreducible representations are self-contained and cannot be decomposed into smaller parts.
The character table of a group is a square matrix that summarizes the properties of the group's irreducible representations. Each row and column in the character table corresponds to a particular irreducible representation, and the values in the table represent how the group's operations transform the basis elements of each representation. By finding the irreducible representations and their corresponding characters, the character table can be constructed and used to analyze the symmetry of a physical system.