  We establish the theoretical framework for an exact description of the open
system dynamics of permutationally invariant (PI) states in arbitrary $N$-qudit
systems when this dynamics preserves the PI symmetry over time. Thanks to
Schur-Weyl duality powerful formalism, we identify an orthonormal operator
basis in the PI operator subspace of the Liouville space onto which the master
equation can be projected and we provide the exact expansion coefficients in
the most general case. Our approach does not require to compute the Schur
transform as it operates directly within the restricted operator subspace,
whose dimension only scales polynomially with the number of qudits. We
introduce the concept of $3\nu$-symbol matrix that proves to be very useful in
this context.
