  We construct an ensemble of correlation matrices from high-frequency foreign
exchange market data, with one matrix for every day for 446 days. The matrices
are symmetric and have vanishing diagonal elements after subtracting the
identity matrix. For this case, we construct the general permutation invariant
Gaussian matrix model, which has 4 parameters characterised using the
representation theory of symmetric groups. The permutation invariant polynomial
functions of the symmetric, diagonally vanishing matrices have a basis labelled
by undirected loop-less graphs. Using the expectation values of the general
linear and quadratic permutation invariant functions of the matrices in the
dataset, the 4 parameters of the matrix model are determined. The model then
predicts the expectation values of the cubic and quartic polynomials. These
predictions are compared to the data to give strong evidence for a good overall
fit of the permutation invariant Gaussian matrix model. The linear, quadratic,
cubic and quartic polynomial functions are then used to define low-dimensional
feature vectors for the days associated to the matrices. These vectors, with
choices informed by the refined structure of small non-Gaussianities, are found
to be effective as a tool for anomaly detection in market states: statistically
significant correlations are established between atypical days as defined using
these feature vectors, and days with significant economic events as recognized
in standard foreign exchange economic calendars. They are also shown to be
useful as a tool for ranking pairs of days in terms of their similarity,
yielding a strongly statistically significant correlation with a ranking based
on a higher dimensional proxy for visual similarity.
