= Solution
Apply the theorem to a symmetric injective MPS. The physically transformed tensor $A_g^i=\sum_jU(g)_{ij}A^j$ generates the same state, so the theorem forces
$$
A_g^i=e^{i\theta(g)}V(g)A^iV(g)^{-1}.
$$
Applying $g$ and then $h$ shows that $V(g)V(h)$ and $V(gh)$ implement the same gauge transformation. Injectivity makes that gauge unique up to a scalar, hence $V(g)V(h)=\omega(g,h)V(gh)$. Associativity gives the two-cocycle equation for $\omega$.
A continuous symmetry-preserving gapped path changes the tensor and $V(g)$ continuously but cannot change the discrete cohomology class $[\omega]$ without losing injectivity, breaking the symmetry, or closing the gap. Thus the <fundamental theorem of matrix product states> turns the virtual projective representation into the invariant classifying one-dimensional <symmetry-protected topological phases>. With broken symmetry, one first records the permuted ground-state sectors and then applies the same argument to their unbroken subgroup.
Back to article page
