IRREDUCIBLE TENSOR BASES FOR THE FROBENIUS ALGEBRA OF A FINITE UNITARY-GROUP

Symmetry adaptation to a finite group is often accomplished by projection with operators that are elements of a matric basis of the Frobenius algebra of the group. The matric basis elements are expressed in terms of the regular, or group, basis of the algebra by means of the irreducible representations. It is shown here for a unitary group that this algebra is also spanned by an irreducible tensor basis expressed on the matric basis with coupling coefficients. The centrum of the algebra is spanned by the invariant elements of this basis. Another irreducible tensor basis is obtained by completely reducing the representation of the group generated by similarity transforming itself. This conjugacy representation is partially reduced by the conjugacy classes so that each of the tensor basis elements is a linear combination of elements from a given class. The relation between this basis and the previous one is discussed. An example is provided for the point group C3v isomorphic to the symmetric group S(3).