  We develop a comprehensive theory for the effective dynamics of Bloch
electrons based on symmetry. We begin with a scheme to systematically derive
the irreducible representations (IRs) characterizing the Bloch functions.
Starting from a tight-binding (TB) approach, we decompose the TB basis
functions into localized symmetry-adapted atomic orbitals and crystal-periodic
symmetry-adapted plane waves. Each of these subproblems is independent of the
details of a particular crystal structure and it is largely independent of the
other subproblem, hence permitting for each subproblem an independent universal
solution. Taking monolayer MoS$_2$ and few-layer graphene as examples, we
tabulate the symmetrized $p$ and $d$ orbitals as well as the symmetrized plane
wave spinors for these systems. The symmetry-adapted basis functions
block-diagonalize the TB Hamiltonian such that each block yields eigenstates
transforming according to one of the IRs of the group of the wave vector $G_k$.
For many crystal structures, it is possible to define multiple distinct
coordinate systems such that for wave vectors $k$ at the border of the
Brillouin zone the IRs characterizing the Bloch states depend on the coordinate
system, i.e., these IRs of $G_k$ are not uniquely determined by the symmetry of
a crystal structure. The different coordinate systems are related by a
coordinate shift that results in a rearrangement of the IRs of $G_k$
characterizing the Bloch states. We illustrate this rearrangement with three
coordinate systems for MoS$_2$ and tri-layer graphene. Using monolayer MoS$_2$
as an example, we combine the symmetry analysis of its bulk Bloch states with
the theory of invariants to construct a generic multiband Hamiltonian for
electrons near the $K$ point of the Brillouin zone. The Hamiltonian includes
the effect of spin-orbit coupling, strain and external electric and magnetic
fields.
