  We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by
investigating their adjacency matrices and their spectra. As a first step, we
characterize switching equivalence and balance of gain graphs in terms of their
adjacency matrices in $M_n(\mathbb C G)$. Then we introduce a represented
adjacency matrix, associated with a gain graph and a group representation, by
extending the theory of Fourier transforms from the group algebra $\mathbb C G$
to the algebra $M_n(\mathbb C G)$. We prove that a gain graph is balanced if
and only if the spectrum of the represented adjacency matrix associated with
any (or equivalently all) faithful unitary representation of $G$ coincides with
the spectrum of the underlying graph, with multiplicity given by the degree of
the representation. We show that the complex adjacency matrix of unit gain
graphs and the adjacency matrix of a cover graph are indeed particular cases of
our construction. This enables us to recover some classical results and prove
some new characterizations of balance in terms of spectrum, index or structure
of these graphs.
