  The Ginzburg-Landau (GL) theory is very successful in describing the pairing
symmetry, a fundamental characterization of the broken symmetries in a paired
superfluid or superconductor. However, GL theory does not describe fermionic
excitations such as Bogoliubov quasiparticles or Andreev bound states that are
directly related to topological properties of the superconductor. In this work,
we show that the symmetries of the fermionic excitations are captured by a
Projective Symmetry Group (PSG), which is a group extension of the bosonic
symmetry group in the superconducting state. We further establish a
correspondence between the pairing symmetry and the fermion PSG. When the
normal and superconducting states share the same spin rotational symmetry,
there is a simpler correspondence between the pairing symmetry and the fermion
PSG, which we enumerate for all 32 crystalline point groups. We also discuss
the general framework for computing PSGs when the spin rotational symmetry is
spontaneously broken in the superconducting state. This PSG formalism leads to
experimental consequences, and as an example, we show how a given pairing
symmetry dictates the classification of topological superconductivity.
