By counting the number of objects that are unchanged or changed by a sign (i.e. rotated by 180o we can determine the character of each symmetry operation in the given basis. Each of the characters can be thought of as a component of the length of a vector in the space of the point group. This vector in the space is called the reducible representation.
We can think of the reducible representation as a vector in a space. The space is given by the point group symmetry. The process of decomposing the reducible representation is actually taking the dot product of the vector with each of the basis vectors (i.e. the irreps) in the space. When we multiply the character of the reducible representation by the irrep that is a dot product. Consider the definition of a dot product:
We see that the dot product is a product in each dimension. In Cartesisn space the dimensions are x, y and z. In a point group space the dimensions are given by the symmetry operations. Thus, the standard formula for determining the projection of the reducible representation on the irreps of the point group is analogous to a dor product. The formula given in the text books is:
In this formula the g(R) is just the number of operations in a class. This is only needed because we write the character table in a compact way that places all of the operations in class in a single column of the table. The factor 1/h is simply the normalization nad we could also take a normalized dot product. The essential aspect of the projections if the multiplication of the character of the irrep by the character of the reducible representation. This aspect is exactly analogous to the multiplication of the components along each Cartesian coordinate in the dor product. In both cases, the procedure has the result of determining the production of one vector on another.
