A representation of a symmetry operation of a group, which cannot be expressed in terms of a representation of lower dimension. When the representation of the group is in matrix form (i.e. a set of matrices that multiply in the same way as the elements of the group), the matrix representation cannot be put into block-diagonal form by constructing a linear combination of the basis functions. The importance of irreducible representations in quantum mechanics is that the energy levels of the system are labelled by the irreducible representations of the symmetry group of the system, thus enabling selection rules to be deduced. In contrast to an irreducible representation, a reducible representation can be expressed in terms of a representation of lower dimension, with a reducible matrix representation that can be put into block diagonal form by constructing a linear combination of the basis functions.