In field theory we associate to each Gauge theory a continuous group of local transformations (a Gauge group), and then we require\define fermion fields to be irreducible representations belonging to the fundamental representation of this Gauge group.
What is the fundamental representation, and why do we require fermions to be in it?
What does it mean for a field to belong to a certain representation? Is this just a way of stating that the fields are the "targets" of the Gauge transformations we introduced, meaning that they belong to the vector space where the representation of these transformations act?
A Gauge transformation must by definition not change the physics. This means that given any field $\psi$ we have to identify as a single physical entity all the fields that can be obtained by $\psi$ by means of any element of the Gauge group. Is the formalization of this concept the reason we require fields to belong to irreducible representations? Is there any other justification for it?