Hidden symmetries and Large N factorisation for permutation invariant matrix observables

TL;DR

This paper explores permutation invariant matrix observables, establishing their correspondence with diagram algebra elements, revealing an enhanced symmetry, and proving a large N factorisation property in certain Gaussian models.

Contribution

It introduces a novel correspondence between permutation invariant observables and partition algebra elements, and proves large N factorisation for these observables in specific models.

Findings

Permutation invariant observables correspond to partition algebra classes.

Enhanced O(N) symmetry exists in a subspace of the model.

Large N factorisation extends to permutation invariant matrix observables.

Abstract

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 of this inner product, which generalizes a familiar large $N$ factorisation for inner products of matrix traces invariant under continuous symmetries.