  We study the different phases and the phase transitions in a system of
$Y$-shaped particles, examples of which include Immunoglobulin-G and
trinaphthylene molecules, on a triangular lattice interacting exclusively
through excluded volume interactions. Each particle consists of a central site
and three of its six nearest neighbours chosen alternately, such that there are
two types of particles which are mirror images of each other. We study the
equilibrium properties of the system using grand canonical Monte Carlo
simulations that implements an algorithm with cluster moves that is able to
equilibrate the system at densities close to full packing. We show that, with
increasing density, the system undergoes two entropy-driven phase transitions
with two broken-symmetry phases. At low densities, the system is in a
disordered phase. As intermediate phases, there is a solid-like sublattice
phase in which one type of particle is preferred over the other and the
particles preferentially occupy one of four sublattices, thus breaking both
particle-symmetry as well as translational invariance. At even higher
densities, the phase is a columnar phase, where the particle-symmetry is
restored, and the particles preferentially occupy even or odd rows along one of
the three directions. This phase has translational order in only one direction,
and breaks rotational invariance. From finite size scaling, we demonstrate that
both the transitions are first order in nature. We also show that the simpler
system with only one type of particles undergoes a single discontinuous phase
transition from a disordered phase to a solid-like sublattice phase with
increasing density of particles.
