  The theory of Wilson loops for gauge theories with unitary gauge groups is
formulated in the language of symmetric functions. The main objects in this
theory are two generating functions, which are related to each other by the
involution that exchanges an irreducible representation with its conjugate.
Both of them contain all information about the Wilson loops in arbitrary
representations as well as the correlators of multiply-wound Wilson loops. This
general framework is combined with the results of the Gaussian matrix model,
which calculates the expectation values of $1/2$-BPS circular Wilson loops in
$\mathcal{N}=4$ Super-Yang-Mills theory. General, explicit, formulas for the
connected correlators of multiply-wound Wilson loops in terms of the traces of
symmetrized matrix products are obtained, as well as their inverses. It is
shown that the generating functions for Wilson loops in mutually conjugate
representations are related by a duality relation whenever they can be
calculated by a Hermitian matrix model.
