We present a method for the exact diagonalization of the SU(N) Heisenberg interaction Hamiltonian using Young tableaus to work directly in each irreducible representation of the global SU(N) group. This generalized scheme is applicable to chains consisting of several particles per site, with any SU(N) symmetry at each site. Extending some of the key results of substitutional analysis, we demonstrate how basis states can be efficiently constructed for the relevant SU(N) subsector, which, especially with increasing values of N or numbers of sites, has a much smaller dimension than the full Hilbert space. This allows us to analyze systems of larger sizes than can be handled by existing techniques. We apply this method to investigate the presence of edge states in SU(N) Heisenberg and Affleck-Kennedy-Lieb-Tasaki Hamiltonians.