  Symmetries impose structure on the Hilbert space of a quantum mechanical
model. The mathematical units of this structure are the irreducible
representations of symmetry groups and I consider how they function as
conceptual units of interpretation. For models with symmetry, the properties of
irreducible representations constrain the possibilities of Hilbert space
arithmetic, i.e.\ how a Hilbert space can be decomposed into sums of subspaces
and factored into products of subspaces. Partitioning the Hilbert space is
equivalent to parsing the system into subsystems, and these emergent subsystems
provide insight into the kinematics, dynamics, and informatics of a quantum
model. This article provides examples of how complex models can be built up
from irreducible representations that correspond to `natural' ontological units
like spins and particles. It also gives examples of the reverse process in
which complex models are partitioned into subsystems that are selected by the
representations of the symmetries and require no underlying ontological
commitments. These techniques are applied to a few-body model in one-dimension
with a Hamiltonian depending on an interaction strength parameter. As this
parameter is tuned, the Hamiltonian runs dynamical spectrum from integrable to
chaotic, and the subsystems relevant for analyzing and interpreting the
dynamics shift accordingly.
