  We present an analytical description for the collective dynamics of
oscillator ensembles with higher-order coupling encoded by simplicial
structure, which serves as an illustrative and insightful paradigm for brain
function and information storage. The novel dynamics of the system, including
abrupt desynchronization and multistability, are rigorously characterized and
the critical points that correspond to a continuum of first-order phase
transitions are found to satisfy universal scaling properties. More
importantly, the underlying bifurcation mechanism giving rise to multiple
clusters with arbitrary ensemble size is characterized using a rigorous
spectral analysis of the stable cluster states. As a consequence of $SO_2$
group symmetry, we show that the continuum of abrupt desynchronization
transitions result from the instability of a collective mode under the
nontrivial antisymmetric manifold in the high dimensional phase space.
