  We present a general symmetry-based framework for obtaining many-body
Hamiltonians with scarred eigenstates that do not obey the eigenstate
thermalization hypothesis. Our models are derived from parent Hamiltonians with
a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as
degenerate multiplets that transform as irreducible representations of the
symmetry (`tunnels'). We show that large classes of perturbations break the
symmetry, but in a manner that preserves a particular low-entanglement
multiplet of states -- thereby giving generic, thermal spectra with a `shadow'
of the broken symmetry in the form of scars. The generators of the Lie algebra
furnish operators with `spectrum generating algebras' that can be used to lift
the degeneracy of the scar states and promote them to equally spaced `towers'.
Our framework applies to several known models with scars, but we also introduce
new models with scars that transform as irreducible representations of
symmetries such as SU(3) and $q$-deformed SU(2), significantly generalizing the
types of systems known to harbor this phenomenon. Additionally, we present new
examples of generalized AKLT models with scar states that do not transform in
an irreducible representation of the relevant symmetry. These are derived from
parent Hamiltonians with enhanced symmetries, and bring AKLT-like models into
our framework.
