Symmetries in zero and finite center-of-mass momenta excitons

We present a symmetry-based framework for the analysis of excitonic states, incorporating both time-reversal and space-group symmetries. We demonstrate the use of time-reversal and space-group symmetries to obtain exciton eigenstates at symmetry-related center-of-mass momenta in the entire Brillouin zone from eigenstates calculated for center-of-mass momenta in the irreducible Brillouin zone. Furthermore, by explicitly calculating the irreducible representations of the little groups, we classify excitons according to their symmetry properties across the Brillouin zone. Using projection operators, we construct symmetry-adapted linear combinations of electron-hole product states, which block-diagonalize the Bethe-Salpeter Equation (BSE) Hamiltonian at both zero and finite exciton center-of-mass momenta. This enables a transparent organization of excitonic states and provides direct access to their degeneracies, selection rules, and symmetry-protected features. As a demonstration, we apply this formalism to monolayer MoS$_{2}$, where the classification of excitonic irreducible representations and the block structure of the BSE Hamiltonian show excellent agreement with compatibility relations derived from group theory. Beyond this material-specific example, the framework offers a general and conceptually rigorous approach to the symmetry classification of excitons, enabling significant reductions in computational cost for optical spectra, exciton-phonon interactions, and excitonic band structure calculations across a wide range of materials.