  The 1-form symmetry, manifesting as loop-like symmetries, has gained
prominence in the study of quantum phases, deepening our understanding of
symmetry. However, the role of 1-form symmetries in Projected Entangled-Pair
States (PEPS), two-dimensional tensor network states, remains largely
underexplored. We present a novel framework for understanding 1-form symmetries
within tensor networks, specifically focusing on the derivation of algebraic
relations for symmetry matrices on the PEPS virtual legs. Our results reveal
that 1-form symmetries impose stringent constraints on tensor network
representations, leading to distinct anomalous braiding phases carried by
symmetry matrices. We demonstrate how these symmetries influence the ground
state and tangent space in PEPS, providing new insights into their physical
implications for enhancing ground state optimization efficiency and
characterizing the 1-form symmetry structure in excited states.
