  The study of band connectivity is a fundamental problem in condensed matter
physics. Here, we develop a new method for analyzing band connectivity, which
completely solves the outstanding questions of the reducibility and
decomposition of band complexes. By translating the symmetry conditions into a
set of band balance equations, we show that all possible band structure
solutions can be described by a positive affine monoid structure, which has a
unique minimal set of generators, called Hilbert basis. We show that Hilbert
basis completely determine whether a band complex is reducible and how it can
be decomposed. The band complexes corresponding to Hilbert basis vectors,
termed as Hilbert band complexes (HBCs), can be regarded as elementary building
blocks of band structures. We develop algorithms to construct HBCs, analyze
their graph features, and merge them into large complexes. We find some
interesting examples, such as HBCs corresponding to complete bipartite graphs,
and complexes which can grow without bound by successively merging a HBC.
