  We study spinless fermions in a flux threaded AB$_2$ chain taking into
account nearest-neighbor Coulomb interactions. The exact diagonalization of the
spinless AB$_2$ chain is presented in the limiting cases of infinite or zero
nearest-neighbor Coulomb repulsion for any filling. Without interactions, the
AB$_2$ chain has a flat band even in the presence of magnetic flux. We show
that the respective localized states can be written in the most compact form as
standing waves in one or two consecutive plaquettes. We show that this result
is easily generalized to other frustrated lattices such as the Lieb lattice. A
restricted Hartree-Fock study of the $V/t$ versus filling phase diagram of the
AB$_2$ chain has also been carried out. The validity of the mean-field approach
is discussed taking into account the exact results in the case of infinite
repulsion. The ground-state energy as a function of filling and interaction $V$
is determined using the mean-field approach and exactly for infinite or zero
$V$. In the strong-coupling limit, two kinds of localized states occur:
one-particle localized states due to geometry and two-particle localized states
due to interaction and geometry. These localized fermions create open boundary
regions for itinerant carriers. At filling $\rho=2/9$ and in order to avoid the
existence of itinerant fermions with positive kinetic energy, phase separation
occurs between a high-density phase ($\rho=2/3$) and a low-density phase
($\rho=2/9$) leading to a metal-insulator transition. The ground-state energy
reflects such phase separation by becoming linear on filling above 2/9. We
argue that for filling near or larger than 2/9, the spectrum of the t-V AB$_2$
chain can be viewed as a mix of the spectra of Luttinger liquids (LL) with
different fillings, boundary conditions, and LL velocities.
