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We present a method for the exact diagonalization of the $\mathrm{SU}(N)$Heisenberg interaction Hamiltonian, using Young tableaux to work directly ineach irreducible representation of the global $\mathrm{SU}(N)$ group. Thisgeneralized scheme is applicable to chains consisting of several particles persite, with any $\mathrm{SU}(N)$ symmetry at each site. Extending some of thekey results of substitutional analysis, we demonstrate how basis states can beefficiently constructed for the relevant $\mathrm{SU}(N)$ subsector, which,especially with increasing values of $N$ or numbers of sites, has a muchsmaller dimension than the full Hilbert space. This allows us to analyzesystems of larger sizes than can be handled by existing techniques. We applythis method to investigate the presence of edge states in $\mathrm{SU}(N)$Heisenberg and AKLT Hamiltonians.

Exact diagonalization of $\mathrm{SU}(N)$ Heisenberg and AKLT chains using the full $\mathrm{SU}(N)$ symmetry