  Permutation invariant Gaussian matrix models were recently developed for
applications in computational linguistics. A 5-parameter family of models was
solved. In this paper, we use a representation theoretic approach to solve the
general 13-parameter Gaussian model, which can be viewed as a zero-dimensional
quantum field theory. We express the two linear and eleven quadratic terms in
the action in terms of representation theoretic parameters. These parameters
are coefficients of simple quadratic expressions in terms of appropriate linear
combinations of the matrix variables transforming in specific irreducible
representations of the symmetric group $S_D$ where $D$ is the size of the
matrices. They allow the identification of constraints which ensure a
convergent Gaussian measure and well-defined expectation values for polynomial
functions of the random matrix at all orders. A graph-theoretic interpretation
is known to allow the enumeration of permutation invariants of matrices at
linear, quadratic and higher orders. We express the expectation values of all
the quadratic graph-basis invariants and a selection of cubic and quartic
invariants in terms of the representation theoretic parameters of the model.
