  This paper explores whether Eguchi-Kawai reduction for gauge theories with
adjoint fermions is valid. The Eguchi-Kawai reduction relates gauge theories in
different numbers of dimensions in the large $N$ limit provided that certain
conditions are met. In principle, this relation opens up the possibility of
learning about the dynamics of 4D gauge theories through techniques only
available in lower dimensions. Dimensional reduction can be understood as a
special case of large $N$ equivalence between theories related by an orbifold
projection. In this work, we focus on the simplest case of dimensional
reduction, relating a 4D gauge theory to a 3D gauge theory via an orbifold
projection. A necessary condition for the large N equivalence between the 4D
and 3D theories to hold is that certain discrete symmetries in the two theories
must not be broken spontaneously. In pure 4D Yang-Mills theory, these
symmetries break spontaneously as the size of one of the spacetime dimensions
shrinks. An analysis of the effect of adjoint fermions on the relevant
symmetries of the 4D theory shows that the fermions help stabilize the
symmetries. We consider the same problem from the point of view of the lower
dimensional 3D theory and find that, surprisingly, adjoint fermions are not
generally enough to stabilize the necessary symmetries of the 3D theory. In
fact, a rich phase diagram arises, with a complicated pattern of symmetry
breaking. We discuss the possible causes and consequences of this finding.
