Finite transformations of SU(4)

The problem of determining the representation matrices of SU(4) is investigated. A convenient set of parameters is first introduced by writing the general element of the group as a product of exponential functions of the generators, and the generators are expressed as differential operators involving these parameters. Special matrix elements of finite transformations with a SU(3) singlet as the initial state are then obtained by solving the eigenvalue equation of the quadratic Casimir operator of SU(4). The solution has the form of a product of elementary functions and threed_mm^j functions of SU(2) and is free from summation over intermediate states. By expanding one of thed_mm^j functions in an appropriate series a sum rule for the special matrix elements of the permutation operator 12343412 is obtained. The discussions are strictly confined to SU(4), but, some of the results given here can be extended to unitary groups of higher dimensions.