A ** reducible representation** is a group representation whose elements either have the same block diagonal matrix form or can undergo similarity transformations with the same invertible matrix to form block diagonal matrices of the same form.

Consider the following representations of the point group:

By inspection, all the elements of have the same block diagonal matrix form of . Therefore, is a reducible representation of the point group.

Let’s consider another representation of the point group:

The elements do not have the same block diagonal matrix form. However, all of them undergo similarity transformations with the same invertible matrix , where and to give . Hence, is also a reducible representation of the point group. Representations that are associated with a similarity transformation are called ** equivalent representations**, i.e. is equivalent to . It is evident that the elements of a reducible representation may not be in the same block diagonal form, and will only have this form if the appropriate basis is chosen.

A final point about reducible representations is that an element of a reducible representation of a group is composed of the direct sum of the matrices of other representations of that correspond to the same element of . For example, of the point group is a result of the direct sum of and . In other words, a reducible representation can be decomposed or reduced to representations of lower dimensions.

An ** irreducible representation** is a group representation whose elements cannot undergo similarity transformations with the same invertible matrix to form block diagonal matrices of the same form. Hence, an irreducible representation cannot be decomposed or reduced further to a representation of lower dimension. and are examples of irreducible representations of the point group. Every point group has a trivial, one-dimensional irreducible representation with each element being 1.