  The sign cancellation between scattering amplitudes makes fermions different
from bosons. We systematically investigate Feynman diagrams' fermionic sign
structure in a representative many-fermion system---a uniform Fermi gas with
Yukawa interaction. We analyze the role of the crossing symmetry and the global
gauge symmetry in the fermionic sign cancellation. The symmetry arguments are
then used to identify the sign-canceled groups of diagrams. Sign-structure
analysis has two applications. Numerically, it leads to a cluster diagrammatic
Monte Carlo algorithm for fast diagram evaluations. The new algorithm is about
$10^5$ times faster than the conventional approaches in the sixth order.
Furthermore, our analysis provides important hints in constructing the relevant
effective field theory for many-fermion systems.
