On two-dimensional tensor network group symmetries

We introduce two-dimensional tensor-network representations of finite groups carrying a 4-cocycle index. We characterize the associated gapped (2+1)D phases that emerge when these anomalous symmetries act on tensor-network ground states. We further develop related tensor-network unitaries that generate symmetric states representing (3+1)D symmetry-protected topological phases. Although aspects of these constructions have been previously addressed, our contribution unifies them within a single tensor-network framework and emphasizes the explicit formulation of local tensor equations encoding global consistency conditions.