We discuss the role of compact symmetry groups, G, in the classification of
gapped ground state phases of quantum spin systems. We consider two
representations of G on infinite subsystems. First, in arbitrary dimensions, we
show that the ground state spaces of models within the same G-symmetric phase
carry equivalent representations of the group for each finite or infinite
sublattice on which they can be defined and on which they remain gapped. This
includes infinite systems with boundaries or with non-trivial topologies.
Second, for two classes of one-dimensional models, by two different methods,
for G=SU(2) in one, and G\subset SU(d), in the other we construct explicitly an
`excess spin' operator that implements rotations of half of the infinite chain
on the GNS Hilbert space of the ground state of the full chain. Since this
operator is constructed as the limit of a sequence of observables, the
representation itself is, in principle, experimentally observable. We claim
that the corresponding unitary representation of G is closely related to the
representation found at the boundary of half-infinite chains. We conclude with
determining the precise relation between the two representations for the class
of frustration-free models with matrix product ground states.