An irreducible representation is a concept from the field of group theory in mathematics and physics. It refers to a way of representing a group, such as a symmetry group, in a vector space such that no proper subspace is invariant under the action of the group. In simpler terms, it means that the representation cannot be broken down into smaller, simpler representations. These representations are crucial in understanding the symmetries of physical systems, particularly in quantum mechanics. For example, the study of quantum states often involves analyzing the irreducible representations of the symmetry groups associated with those states, helping to reveal fundamental properties of particles and their interactions.