  We study the null space degeneracy of open quantum systems with multiple
non-Abelian, strong symmetries. By decomposing the Hilbert space representation
of these symmetries into an irreducible representation involving the direct sum
of multiple, commuting, invariant subspaces we derive a tight lower bound for
the stationary state degeneracy. We apply these results within the context of
open quantum many-body systems, presenting three illustrative examples: a
fully-connected quantum network, the XXX Heisenberg model and the Hubbard
model. We find that the derived bound, which scales at least cubically in the
system size the $SU(2)$ symmetric cases, is often saturated. Moreover, our work
provides a theory for the systematic block-decomposition of a Liouvillian with
non-Abelian symmetries, reducing the computational difficulty involved in
diagonalising these objects and exposing a natural, physical structure to the
steady states - which we observe in our examples.
