  Exact diagonalization and other numerical studies of quantum spin systems are
notoriously limited by the exponential growth of the Hilbert space dimension
with system size. A common and well-known practice to reduce this increasing
computational effort is to take advantage of the translational symmetry $C_N$
in periodic systems. This represents a rather simple yet elegant application of
the group theoretical symmetry projection operator technique. For isotropic
exchange interactions, the spin-rotational symmetry SU(2) can be used, where
the Hamiltonian matrix is block-structured according to the total spin- and
magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of
irreducible tensor operators allows for an efficient and highly parallelizable
implementation to calculate its matrix elements recursively in the
spin-coupling basis. When combining both $C_N$ and SU(2), mathematically, the
symmetry projection technique leads to ready-to-use formulas. However, the
evaluation of these formulas is very demanding in both computation time and
memory consumption, problems which are said to outweigh the benefits of the
symmetry reduced matrix shape. We show a way to minimize the computational
effort for selected systems and present the largest numerically accessible
cases.
