Hidden symmetries and large N factorisation for permutation invariant matrix observables

Abstract

Permutation invariant polynomial functions of matrices have previously been studied as the observables in matrix models invariant under SN, 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 Pk (N). On a 4-dimensional subspace of the 13-parameter space of SN 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 Pk (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.