  The high-$T_c$ cuprates are possible candidates for d-wave superconductivity,
with the Cooper pair wave function belonging to a non-trivial irreducible
representation of the lattice point group. We argue that this d-wave symmetry
is related to a special form of the fermionic kinetic energy and does not
require any novel pairing mechanism. In this context, we present a detailed
study of the bound states and resonances formed by two lattice fermions
interacting via a non-retarded potential that is attractive for nearest
neighbors but repulsive for other relative positions. In the case of strong
binding, a pair formed by fermions on adjacent lattice sites can have a small
effective mass, thereby implying a high condensation temperature. For a weakly
bound state, a pair with non-trivial symmetry tends to be smaller in size than
an s-wave pair. These and other findings are discussed in connection with the
properties of high-$T_c$ cuprate superconductors.
