  We present a class of Hamiltonians $H$ for which a sector of the Hilbert
space invariant under a Lie group $G$, which is not a symmetry of $H$,
possesses the essential properties of many-body scar states. These include the
absence of thermalization and the "revivals" of special initial states in time
evolution. Some of the scar states found in earlier work may be viewed as
special cases of our construction. A particular class of examples concerns
interacting spin-1/2 fermions on a lattice consisting of $N$ sites (it includes
deformations of the Fermi-Hubbard model as special cases), and we show that it
contains two families of $N+1$ scar states. One of these families, which was
found in recent literature, is comprised of the well-known $\eta$-pairing
states. We find another family of scar states which is $U(N)$ invariant. Both
families and most of the group-invariant scar states produced by our
construction in general, give rise to the off-diagonal long range order which
survives at high temperatures and is insensitive to the details of the
dynamics. Such states could be used for reliable quantum information processing
because the information is stored non-locally, and thus cannot be easily erased
by local perturbations. In contrast, other scar states we find are product
states which could be easily prepared experimentally. The dimension of scar
subspace is directly controlled by the choice of group $G$ and can be made
exponentially large.
