 Solution

For a unique gapped ground state, one-dimensional bosonic phases protected by an on-site symmetry $G$ are classified by the cohomology classof a projective virtual symmetry of a matrix product state. After blocking finitely many sites, an injective matrix product state tensor satisfiesChanging the virtual gauge or rephasing $V(g)$ changes $\omega$ by a coboundary and leaves $[\omega ]$ fixed.

If symmetry breaking and degenerate ground states are allowed, the additional data are the unbroken subgroup $H\subseteq G$, the way $G$ permutes the broken-symmetry sectors, and a class in $H^2(H,U(1))$ within one sector. Acting with ${⨂}_i{U}_i(g)$ preserves a symmetric ground state up to phase in the unbroken case; in a broken phase it generally permutes the different ground states.