  We develop a framework that provides a straightforward approach to fully
exploit the permutational symmetry of identical multi-level systems. By taking
into account the permutational symmetry, we outline a simple scheme that allows
to map the dynamics of $N$ identical $d$-level systems to the dynamics of $d$
bosonic modes with $N$ particles, achieving an exponential reduction on the
dimensionality of the problem in a simple and straightforward way. In
particular, we consider the Lindblad dynamics of several identical multi-level
systems interacting with a common subsystem under the action of collective
dissipation terms.
