  When a quantum field theory in $d$-spacetime dimensions possesses a global
$(d-1)$-form symmetry, it can decompose into disjoint unions of other theories.
This is reflected in the physical quantities of the theory and can be used to
study properties of the constituent theories. In this note we highlight the
equivalence between the decomposition of orbifold $\sigma$-models and
disconnected McKay quivers. Specifically, we show in numerous examples that
each component of a McKay quiver can be given definitive geometric meaning
through the decomposition formulae. In addition, we give a purely group and
representation theoretic derivation of the quivers for the cases where the
trivially acting part of the orbifold group is central. As expected, the
resulting quivers are compatible with the case of $\sigma$-models on `banded'
gerbes.
