Hidden symmetries and large N factorisation for permutation invariant matrix observables

A bstract Permutation invariant polynomial functions of matrices have previously been studied as the observables in matrix models invariant under S N , the symmetric group of all permutations of N objects. In this paper, the permutation invariant matrix observables (PIMOs) of degree k are shown to be in one-to-one correspondence with equivalence classes of elements in the diagrammatic partition algebra P k ( N ). On a 4-dimensional subspace of the 13-parameter space of S N invariant Gaussian models, there is an enhanced O ( N ) symmetry. At a special point in this subspace, is the simplest O ( N ) invariant action. This is used to define an inner product on the PIMOs which is expressible as a trace of a product of elements in the partition algebra. The diagram algebra P k ( N ) is used to prove the large N factorisation property for this inner product, which generalizes a familiar large N factorisation for inner products of matrix traces invariant under continuous symmetries.