Hidden symmetries and Large N factorisation for permutation invariant matrix observables

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.


Introduction
The simplifications of large N in matrix quantum field theories in diverse dimensions with continuous gauge symmetries such as U (N ), O(N ), Sp(N ) discovered in [1] have played a major role in the development of gauge-string duality in subsequent years. This includes low-dimensional non-critical string theories dual to zero-dimensional QFTs (matrix models) [2][3][4], the string dual of two-dimensional Yang-Mills theories [5], and the generalization to higher dimensions in the AdS/CFT correspondence [6]. In theories with continuous symmetries containing adjoint fields, the space of gauge invariants is generated by traces of matrices. An important aspect of simplicity in the large N limit is "large N factorisation". In the context of AdS/CFT, large N factorisation for two-point functions involving gauge invariants built from a complex matrix is an expression of orthogonality for distinct trace structures [7]. This plays an important role in the connection between multi-traces constructed from a small number of matrices and perturbative gravitons in the AdS dual [8,9]. The breakdown of this orthogonality when the number of matrices becomes comparable to N guides the identification of CFT duals [7,10,11] for giant gravitons [12][13][14]. Large N factorisation also enters the construction of gauge-string duals in collective field theory [15][16][17][18], which gives useful insights into the emergence of classical limits at large N . It is also employed in the Master field approach to large N [19] and loop equations [20] (see for example [21][22][23] for recent developments of these themes). In the geometrical construction of gauge-string duality, based on Schur-Weyl duality and branched covers, in instances such as large N 2d Yang Mills [5,[24][25][26][27][28][29][30][31] or the simple toy model of Gaussian Hermitian matrix theory [32][33][34][35], trace structures of matrix invariants correspond to branching structures of covering maps from string worldsheets.
In this paper, we will develop the theme of large N factorisation for permutation invariant matrix models [36][37][38][39]. Permutation invariance was motivated in [36] in the context of matrix data arising in computational linguistics [40][41][42][43]. The formulation of large N factorisation we will use is similar to the one in [7]. We will use the simplest inner product on the space of permutation invariant matrix observables (PIMOs). It comes from a special point on the moduli space of S N invariant Gaussian matrix models where the action has O(N ) symmetry. This is the first sense in which hidden symmetries appear in this paper. Permutation invariant random matrix distributions have also been studied from the point of view of mathematical statistics, using partition algebra diagrams [44][45][46].
The second kind of hidden symmetry appearing in this paper is based on Schur-Weyl duality. Observables invariant under the action of a symmetry group G, as we will discuss in this paper, are organized by algebras dual to G. For the case of U (N ) symmetry the dual algebras are based on the standard Schur-Weyl duality [47] between U (N ) and S k in the k-fold tensor product of V ⊗k of the fundamental representation V of U (N ). Applications of Schur-Weyl duality to the computation of correlators in matrix models with U (N ) symmetry are developed in [10,[48][49][50][51][52][53][54][55][56][57][58][59][60] and short reviews are [61,62]. The U (N ) case serves as a powerful source of useful analogies throughout the paper. When U (N ) is replaced by S N as the invariance of interest, the Schur-Weyl dual algebras are diagrammatic partition algebras P k (N ). These algebras have been studied in statistical physics and representation theory [63][64][65]. The algebras P k (N ) are finite dimensional and have a basis which can be labelled by diagrams, corresponding to set partitions of a set of 2k objects. A set partition of a set S is a collection of non-empty subsets of S, such that any pair of of subsets has zero intersection, and the union of the subsets is the set S. Equivalently, every element of the set is included in exactly one of the subsets (see for example [66] for further information on set partitions).
The paper is organised as follows. In section 2 we review the construction of the permutation invariant Gaussian 1-matrix model, and the counting of invariant matrix observables developed in [36,37]. Here we give a new description of the counting, which emphasizes the underlying hidden partition algebra symmetry arising as a consequence of Schur-Weyl duality. We end the section with a derivation of the O(N ) symmetric point in the moduli space of S N invariant 1-matrix models.
Section 3 is dedicated to the construction of PIMOs by means of partition algebras. We give a brief description of the partition algebras. In particular we present the diagram basis and describe how the product is computed by using a composition of diagrams. The construction of U (N ) invariants using symmetric group algebras is reviewed as a warmup exercise. This is generalised to give a map from partition algebra elements to PIMOs (equation (3.14)), leading to a correspondence between PIMOs and equivalence classes of partition algebra elements. These equivalence classes are defined in equation (3.17). The simplest O(N ) invariant action is used to define an inner product on the space of PIMOs in terms of a trace of partition algebra elements (equation (3.19)). Section 4 proves the large N factorization of the inner product on PIMOs thus defined. That is, we show that whereÔ i ,Ô j are normalized PIMOs labelled by indices i, j running over equivalence classes of partition algebra elements. The proof of large N factorization relies on the existence of a partial ordering on the diagram basis for the partition algebra. The partial ordering is related to an inclusion of diagrams, and can itself be described by another diagram of diagrams called a Hasse diagram [67]. We end the section by extending the proof to multimatrix observables.

Hidden symmetries in permutation invariant Gaussian matrix models
In [37] a 13-parameter family of Gaussian matrix models consistent with permutation invariance was constructed, by using a transformation from the matrix variables M ij to variables labelled by irreducible representations of S D . The expectation values of linear and quadratic permutation invariant polynomials in M ij were given in terms of the representation theoretic parameters. Expectation values for a sample of cubic and quartic invariant polynomials were constructed using Wick's theorem. Additional examples were computed in [38]. The results were generalized to the 2-matrix case in [39]. Computer code for expectation values of invariant polynomials in the 1-matrix and 2-matrix case is available as part of [39]. The schematic form of the permutation invariant Gaussian matrix model (PIGMM) is The action S(M ) contains two linear terms: L 1 , L 2 ; and eleven quadratic terms Q 1 , . . . Q 11 . It is the most general quadratic action invariant under the following group action of S N (the symmetric group on N objects), The permutations σ ∈ S N are invertible maps σ : {1, · · · , N } → {1, . . . , N }. The product of two permutations σ 1 , σ 2 is defined by composing the maps : σ 1 σ 2 (i) = σ 2 (σ 1 (i)). As an example, consider the following two permutations in S 3 : In this case Let V N be an N -dimensional vector space with orthonormal basis e i , i = 1, . . . , N . The defining representation ρ N : S N → GL(V N ) of S N assigns the following linear operator to every permutation ρ N (σ)e i = e σ −1 (i) . (2.5) From equation (2.2), we see that the vector space spanned by M ij is acted on by S N in the same way as V N ⊗ V N . We have the identification (2.6) This is not an irreducible representation, it decomposes into several irreducible components Here V [N ] corresponds to the trivial one-dimensional representation of S N . The repre- ,2] are non-trivial irreducible representations, labeled by integer partitions of N . Their dimensions are respectively We will use the index Λ 1 ranging over the labels for irreducible representations and we refer to the corresponding irreducible representations of S N as V S N Λ 1 . The above decomposition can be deduced using together with the tensor product rule described in section 7.13 of [68]. See also [69] for a dedicated treatment of symmetric group representation theory. Note that the multiplicity of V [N ] in (2.7) is exactly why there are two linear terms L 1 , L 2 in the action (2.1). Furthermore the isomorphism in equation (2.7) implies that there exists a set of linear combinations of matrix elements labelled by Λ 1 The index a is a state index for the irreps, while α is a multiplicity index The change of basis is given by the Clebsch-Gordan coefficients C Λ 1 ,α a,ij . They have the property where the matrices D Λ 1 ab (σ) are irreducible matrix representations of S N (background on the Clebsch-Gordan coefficients for symmetric groups is available in [68]). Without loss of generality, we can assume that the Clebsch-Gordan coefficients define an orthonormal basis with respect to the inner product (2.14) Equivalently, the representation theoretic variables satisfy Together with the fact that the inner product (2.14) is S N invariant, it follows that Using the above basis it immediately follows that the quadratic combination is an invariant polynomial, where A useful observation is that, while the Clebsch-Gordan coefficients depend on a choice of basis for every irreducible component in (2.7), the tensors Q Λ 1 ,αβ ijkl do not. For all four Λ 1 , they can be constructed by using only the explicit bases for the subspaces V [N ] and V [N −1,1] in (2.7) [37].
We may associate a unique parameter to every invariant. Since there is a symmetric matrix of dimension Mult(V N ⊗ V N → V S N Λ 1 ) parametrising the quadratic part of the action, for every choice of Λ 1 . Using the multiplicities in the decomposition (2.7), we have independent parameters. The two linear terms are given by The quadratic part is where the matrices g Λ 1 αβ are parameters of the model. In this basis the partition function is, (2.23) The matrices g Λ 1 αβ must have non-negative eigenvalues to define a convergent integral. Note that the parameters in the quadratic part of the action in [37] are related to the parameters in this paper as αβ , (2.24) The slight shift of notation makes the connection between the parameters and the decomposition (2.7) more manifest. These permutation invariant matrix observables (PIMOs) can be organized by their degree. At degree k, the matrix monomials

Counting matrix observables using partition algebras
form a basis for a vector space isomorphic to Sym k (V N ⊗ V N ). The symmetric group S k acts on (V N ⊗ V N ) ⊗k by permuting the k tensor factors. The subspace Sym k (V N ⊗ V N ) is the subspace of S k invariants in (V N ⊗ V N ) ⊗k . This S k invariance is imposed because of the bosonic symmetry of the matrix variables M ij . The PIMOs form the Note that the action of τ ∈ S k on (V N ⊗ V N ) ⊗k commutes with the action of σ ∈ S N . This follows since the same σ is applied to all tensor factors.
In [36] the dimension of the space of independent PIMOs for matrices of size N and polynomial degree k was obtained as The initial sums run over integer partitions (Young diagrams) p of N , and integer partitions q of k while the final sum is over the integer divisors l of i. The equation (2.28) computes the multiplicity of the trivial representation of S N ×S k in the decomposition of ( , then the dimension of S N × S k invariants is given by The generalization to multi-matrix observables and a proof of their correspondence with colored directed graphs was developed in [39]. The approach in this paper is based on a new way of counting PIMOs, utilising the connection between dual algebras and matrix invariants. We begin by reviewing this connection in the case of U (N ) invariants. Tensor products of the defining representation V of U (N ) have a multiplicity free decomposition into irreducible representations of U (N ) × S k labelled by Young diagrams (2.31) The sum runs over Young diagrams Λ with k boxes, and for k > N is restricted such that the number of rows l(Λ) in the Young diagram Λ is not greater than N . In the remainder of this paper we will assume N ≥ k for discussions of the unitary group. This result is known as Schur-Weyl duality (see chapter 6 in [47]). On the left-hand side of this equation we have a basis e i 1 ⊗ e i 2 ⊗ · · · ⊗ e i k with each index i running from 1 to N . On the right-hand side For a fixed Young diagram Λ and a fixed state M in V (2.33) It is well-known that U (N ) invariant matrix observables have a basis of multi-traces. These traces can be parameterised by conjugacy classes of permutations. A description of the connection between gauge invariant observables and equivalence classes of permutations for single matrix as well as multi-matrix problems, with applications to AdS/CFT is given in [62]. We review the connection here with an emphasis on Schur-Weyl duality from the outset. This framework, as explained in [62], can be used to give a description of finite N effects on the counting and construction of gauge invariant observables, but we will focus here, as previously mentioned, on the case N ≥ k. For the unitary group the matrix elements M ij are isomorphic to V ⊗ V * , where V * is the complex conjugate representation of V . In other words, U ∈ U (N ) acts on M by conjugation, where the second line follows from Schur's Lemma which implies equation (2.36). Since we are looking for U (N ) invariant polynomials of degree k in M ij , the counting is given by the (2.38) Thus the counting of U (N ) invariants is controlled by the symmetric group algebra, which appeared through Schur-Weyl duality.
Similarly in the case of S N invariant observables there is a dual algebra at play. The dual algebra for the defining representation of S N is called the partition algebra, denoted P k (N ) [63,64]. The representations of the partition algebra determine the multiplicities of S N irreducible representations in the decomposition (see section 2.5 in [70]) . (2.39) The , which is an integer partition of N , is constructed by placing the diagram Λ # 1 below a first row of N − l boxes. Requiring Λ 1 to be a valid Young diagram imposes some constraints on Λ # 1 , which are not manifest in (2.39). This occurs for N < 2k as we explain, while it does not occur for N ≥ 2k. The latter is called the stable limit. To understand this, we denote the first row length of Λ # 1 by (2.40) The inequality follows since l ≤ k in equation (2.39). We also have This follows because Λ # 1 has no more than k boxes.
This is non-trivial condition since Λ # 1 can have up to k boxes. Note that the symmetric group algebra CS k is a subalgebra of P k (N ) (permutations of the tensor factors commute with the action of S N on V ⊗k N ). We can restrict any representation V to CS k to give a decomposition of the form There is a single S N invariant state in every tensor product and considering the dimension of this subspace of (2.47) The sum of squares is indicative of a matrix (Artin-Wedderburn) decomposition [71,72] of a hidden algebra parametrising PIMOs (we found the exposition of the Artin-Wedderburn decomposition in [73] to be useful). We will turn to an explicit construction of PIMOs using partition algebra elements in line with the counting (2.47) in section 3. This sum of squares form in counting invariants, and their connection to the Artin-Wedderburn structure of algebras, have been used in a number of multi-matrix and tensor model applications, e.g. [74][75][76][77][78].

Enhanced O(N) symmetry in parameter space
The In this model, the matrix elements are not statistically independent, but the linear and quadratic moments are readily solvable, as we now show. Higher moments can be obtained using Wick's theorem. This 4-parameter family is a special case of recently studied [36][37][38][39] more general Gaussian matrix models, with permutation symmetry. We now solve for the second moments of matrix variables for the model in (2.49) and compare with the second moments for the permutation invariant Gaussian 1-matrix model. This gives a system of linear equations for the parameters in the permutation invariant Gaussian 1-matrix model in terms of the parameters α, β, γ. See appendix A for an algorithm and computer code to reproduce these results.
We begin by rewriting the action: then the action can be expressed as The vector µ is with the first N terms equal to ǫ and the rest 0 and (2.54) The inverse of G 2 is while the inverse of G 1 is given by From the form of these inverse matrices we can write down the connected two-point function and collecting like terms we are left with the following expression for the two-point function The parameters a, b, c, d satisfy a + b + d = c and therefore the fully simplified two-point function is given by Comparing this to the two-point function of the permutation invariant matrix model (equation (3.6) in [37]) we find that it is reproduced in following parameter limit where we have again written g instead of Λ for our quadratic couplings, labelling them using integer partitions under the identification given in (2.24).
There is a special point in this limit that recovers the two-point as the only non-zero parameters. A quick check on the above computation is the following. Using Clebsch-Gordan coefficients we have where the second line uses orthogonality of the Clebsch-Gordan coefficients. Comparing with equation (2.23) recovers the parameter limit (2.63). 1

Permutation Invariant Matrix Observables (PIMOs)
We will now describe the partition algebra and how the PIMOs are constructed from the S k invariant subalgebra of P k (N ). Properties of the partition algebra [63][64][65]80] will allow us to prove large N factorisation of PIMOs in the O(N ) symmetric matrix model.
The partition algebra P k (N ) is a diagram algebra. It has a finite basis, labelled by diagrams, where multiplication is a type of composition of diagrams. A diagram in P k (N ) has 2k labelled vertices arranged into two rows, with k vertices in each row. Any set of edges between the vertices are allowed. We use the convention in which the bottom vertices are labelled (from left to right) by 1, . . . , k and the top vertices by 1 ′ , . . . , k ′ . For example, P 2 (N ) has a basis of 15 diagrams. Among these are, In general, the dimension of P k (N ) is the number of set partitions of 2k (also known as Bell numbers). The underlying set for this basis of the partition algebra is the set of set partitions of the 2k labelled vertices. There is a redundancy in the diagram picture. The redundancy arises from the fact that we are free to choose any set of edges, as long as every vertex in a subset of the set partition can be reached from any other vertex in the same subset, by a path along the edges. For example, the following pair of diagrams correspond to the same element.
= . (3. 3) The product in P k (N ) is independent of the choice of representative diagram. The subset of diagrams with k edges, each connecting a vertex at the top to a vertex at the bottom and where every vertex has exactly one incident edge, span a subalgebra. This subalgebra is isomorphic to the symmetric group algebra CS k . For example, there is a one-to-one correspondence between permutations in S 3 and the following set of diagrams, , , (3.5)

Construction of PIMOs
We will construct degree k PIMOs from elements d ∈ P k (N ). As a warm-up, we recap the construction of U (N ) invariants using elements in CS k . See [62] for a review of the background literature. For this construction it will be useful to rewrite M ij as M i j and think of these as the matrix elements of a linear operator acting on V , the defining representation of U (N ). Define M to be the linear operator M : V → V with matrix elements in a basis e i for V . In diagram notation the linear operator M is represented by a box labelled M , with one incoming and one outgoing edge, The operator M ⊗k acts on V ⊗k as M ⊗k e i 1 ⊗ · · · ⊗ e i k = M e i 1 ⊗ · · · ⊗ M e i k . (3.8) Diagrammatically, tensor products of operators are represented by horizontally composing the diagrams, When viewed as a matrix polynomial, the trace is a unitary invariant of degree k. The matrix elements of the permutation τ as a linear operator on V ⊗k are (τ ) The diagram representing τ is obtained by associating an edge with every Kronecker delta. For example, for τ = (12) we have the diagram The horizontal lines in equation (3.10) are used to indicate that the incoming and outgoing edges are identified, as expected from a trace. Invariance under the action of U (N ) follows because τ ∈ S k commutes with any U (N ) acting on V ⊗k . The correspondence between gauge invariant operators and permutations has a redundancy given by, This follows because γ −1 M ⊗k γ = M ⊗k . Therefore, a basis of multi-trace observables is in one-to-one correspondence with conjugacy classes of S k , as expected from the counting in equation (2.38). The construction of degree k PIMOs from elements of P k (N ) is identical. For any d ∈ P k (N ), the matrix polynomial is a PIMO, because d commutes with the action of S N acting on V ⊗k N . The matrix elements (d) ..i k also correspond to the diagram representation by associating every Kronecker delta to an edge connecting a pair of vertices. For example, As before, for any γ ∈ S k we have Degree k PIMOs are in one-to-one correspondence with the S k invariant subalgebra of P k (N ). A basis is given by the set of distinct equivalence classes

Inner product on PIMOs
The simplest O(N ) invariant matrix model has the quadratic expectation value  (3.20) where the Kronecker deltas have been replaced by edges. The two-point function in equation (3.19) can be represented by the diagram in the first line below The second line is the sum over Wick contractions parameterized by γ ∈ S k . The last equality comes from straightening the diagram. By following the lines and recording the operators encountered on the way, we recognize the last diagram as the representation of . The symmetry of the two-point function is proved by observing that We have used the invariance of the trace under transposition, cyclicity of the trace and a relabelling of γ → γ −1 . The non-degeneracy of the two-point function at large N follows from the factorization property in the next section. The non-degeneracy at all orders in 1/ √ N is proved in the companion paper by exhibiting an orthogonal basis constructed using representation theory data [81]. This shows that the two-point function defines an inner product.

Large N factorisation
In this section, we will show that the normalized PIMOŝ To prove large N factorization we will study the powers of N appearing in or equivalently, the RHS of equation (3.19) for the two-point function.
It is useful to consider the simpler case This trace can be computed in terms of the number of connected components in the diagram d 1 ∨ d 2 , given by a diagram with all the edges of d 1 and d 2 . In the mathematics literature, this operation is called the join on the partition lattice (see [67]). It is given by To illustrate equation (4.5) consider the following pair of diagrams The join is given by The diagram multiplication gives while the corresponding expression using the join gives To prove this, recall that every edge in a diagram corresponds to a Kronecker delta when acting on V ⊗k N (see examples in (3.15)). Consequently It follows that (4.13) Equivalently, the diagrammatic representation of a trace identifies the bottom vertices with the top vertices, (4.14) Taken literally, this means that we identify the bottom vertices of d T 2 with the top vertices of d 1 , and the top vertices of d T 2 with the bottom vertices of d 1 . The diagram constructed in this manner has all the edges of d 1 together with all the edges of d 2 , which is precisely equal to d 1 ∨ d 2 . See figure 1 for an illustration.
To complete the proof we show that Let b 1 , . . . , b l be sets containing the vertices of connected components of d 1 ∨ d 2 . Then, where the sums over connected components correspond to sums where the indices in each component are set equal. For example,

Factorization for trace form on P k (N)
The proof of the following version of factorization contains most of the essential ingredients necessary for the 1-matrix case. This is a useful warm-up exercise and, as we will see in section 4.2, a special case of factorization in multimatrix models. This equation (4.18) is related to the properties of the distance function defined in proposition 3.1 of [44]. 2 The factorization in equation (4.18) is a consequence of the following where we have used c( We will prove (4.19) by separating the general pairs d 1 , d 2 into three distinct cases: 1. If d 1 only contains edges that are also contained in d 2 , but d 1 = d 2 , we write d 1 < d 2 .
For example, < , and < . (4.20) In this case, d 1 ∨ d 2 = d 2 and it follows that, Note that d 1 < d 2 implies c(d 1 ) > c(d 2 ). Therefore, In this incomparable case, we have since the forming of the join involves adding to d 1 , additional edges creating connections which did not exist in d 1 , or alternatively adding to d 2 additional edges that did not exist in d 2 . Consequently we have the inequality , (4.26) and therefore, As a corollary of the above discussion, which will be useful in the next sub-section, note that if we consider a fixed diagram d 1 and a family of diagrams d 3 with fixed c(d 3 ) such that c(d 1 ) > c(d 3 ), then we have for each d 3 in the family one of the following This follows from (4.21) and (4.24).

Factorization for PIMOs
The 1-matrix connected two-point function (3.19) includes a sum over γ ∈ S k , . (4.29) Large N factorization of PIMOs follows from the inequalities 2 max (4.30) The first step in proving equation (4.30) is to simplify the terms on the RHS. The inequalities in equation (4.19) imply that c(d ∨ γdγ −1 ) is maximized when d = γdγ −1 . Of course, the identity permutation always satisfies this equality. Therefore, (4.31) We are left with proving We employ the same strategy as before, and consider the three distinct cases.
1. Suppose c(d 1 ) > c(d 2 ), and consider the diagrams γd 2 γ −1 for γ ∈ S k . We have c(d 1 ) > c(γd 2 γ −1 ) = c(d 2 ). Assume d 1 , d 2 are such there exists some γ * such that d 1 < γ * d 2 (γ * ) −1 . For any such γ * , the equality in (4.28) implies that Any γ not satisfying this condition leads to d 1 γd 2 γ −1 , and the inequality in (4.28) implies that This implies that The pair is an example of this case since < = . (4.37) The argument is identical for the case where c(d 1 ) < c(d 2 ), and there exists some 2. Secondly, consider the case of incomparability, Recall that for incomparable diagrams (4.25), where the last equality follows because conjugation by a permutation does not change the number of connected components. Therefore (4.42) in this case as well.
3. When d 1 = γd 2 γ −1 for some γ ∈ S k , the bound is saturated and The condition We have proven the inequalities in equation (4.30). As a consequence, we have large N factorization of permutation invariant matrix observables.

Factorization for multi-matrix observables
The above argument generalizes to multi-matrix models. Let M (f ) be n matrices with flavour label f = 1, . . . , n and second moment Permutation invariant multi-matrix observables of degree k = k 1 + k 2 + · · · + k n , where k f is the degree of matrix M (f ) , are constructed using partition algebra elements. Multi-matrix observables are labelled by k = (k 1 , . . . , k n ) and d ∈ P k (N ) As before, we have bosonic symmetry. For any γ ∈ S k ≡ S k 1 × · · · × S kn observables are invariant (4.46) Multi-matrix observables are in one-to-one correspondence with partition algebra equivalence classes Wick contractions vanish unless the flavour indices match, and the sum over γ ∈ S k reduces to a sum over γ ∈ S k (4.48) The same argument holds for the inequality To summarize we have for permutation invariant multi-matrix observables in the above Gaussian O(N ) model. Note that in the case n = k, k f = 1 (all matrices distinct), we have (4.52) Therefore, the sum over Wick contractions reduces to a single element (the identity element). The corresponding two-point function is the first case we considered (equation (4.18)). Finally, we observe that the proof of factorization presented here for general observables labelled by partition algebra diagrams specializes to a new way of thinking about factorization in the case of matrix invariants with continuous symmetry, where the partition algebra diagrams specialize to permutations. The previously known proof based on permutation products can be understood, in the one-matrix case, from the equation This equation is derived and explained as eqn. (2.12) in [62] (multi-matrix generalisations are discussed in references therein). Gauge invariant operators are labelled by permutations σ 1 , σ 2 in conjugacy classes T 1 , T 2 , while |T 1 |, |T 2 | are the sizes of these conjugacy classes.
Large N factorisation follows from the fact that the largest power of N comes from the case where σ 3 is the identity and this only occurs when T 1 = T 2 . In the present way of looking at permutations as special cases of partition algebra diagrams, permutations belonging to distinct conjugacy classes are always incomparable in the partial order on set partitions associated to the diagrams. This corresponds to Case 2 in of the proofs in sections 4.1 and 4.2.

Discussion
In this paper we considered S N invariant matrix models. These can be viewed as generalisations of their more familiar cousins invariant under continuous symmetries. The most general S N invariant Gaussian matrix model is specified by a 13-dimensional parameter space. We have shown that there exists a four-dimensional subspace of the 13-dimensional parameter space in which the S N symmetry is enhanced to O(N ). The parameter limit in which this enhancement takes place is given by equation ( The factorisation property of multi-trace matrix observables invariant under continuous symmetries such as U (N ) in the large N limit is a well known result. We have shown that this continues to hold for S N invariant observables. In the U (N ) case this can be seen using properties of the symmetric group by exploiting the Schur-Weyl duality of U (N ) and S k in order to establish a correspondence between observables and conjugacy classes of S k . Analogously, we gave a description of the permutation invariant matrix polynomial functions in terms of a diagram basis for partition algebras. We used the inner product on the permutation invariant polynomials arising from the simplest O(N ) invariant action, and proved large N factorisation. The partial order on the diagram basis elements, which can itself be described by a Hasse diagram, plays a role in the proof of factorisation.
As explained in the introduction, there are two guiding principles in this paper: the analogies between results for U (N ) invariant matrix models and S N invariant models, and the Schur-Weyl dual algebras of these respective invariances. These principles can be exploited in a number of natural generalisations of the results in this paper. For example, they are applicable to one-dimensional quantum mechanics models of matrices (see our companion paper [81]). They are also applicable to tensor models: this is being developed in [82]. Permutation invariant random matrix distributions have been considered using techniques from probability theory [83]. It would be interesting to investigate the implications of the factorisation results presented here in that context. The 1/N expansion of simple correlators in U (N ) invariant matrix models has a geometrical interpretation in terms of Belyi maps, which are branched covers of the sphere with three branch points [32,33]. This has an interpretation within topological A-model strings [34,35]. The links with tau functions of integrable models are developed in [84]. Matrix model formulations of general Hurwitz space problems are developed in [85]. The present paper shows that the large N simplicity of the trace structures of U (N ) theories extends to the large N simplicity of permutation invariant observables. This suggests that there may well be a rich analogous geometrical story in the large N expansion of permutation invariant models. U (N ) invariant models are related to two-dimensional topological field theories based on lattice gauge theories constructed from symmetric group algebras [54,86]. We expect analogous developments for S N invariant models involving topological field theories based on partition algebras. The partition functions of U (N ) matrix models display rich large N phase structures which should have interesting parallels in the S N invariant case [87][88][89][90][91][92][93][94][95][96]. A recent study of S N lattice gauge theory partition functions is in [97].
Many results in U (N ), SO(N ), Sp(N ) matrix models have been developed in the physical context of gauge-string duality. A natural question which encompasses many of the above technical directions is whether there is a gauge-string dual interpretation for the correlators of permutation invariant observables in the simplest O(N ) invariant model, where we have established large N factorisation. This permutation invariant sector is one which goes beyond singlets of the continuous symmetry. Non-singlet sectors have been organised according to more general representations of the continuous symmetry and discussed in gauge-string duality in connection with low dimensional models of stringy black hole physics [98][99][100]. It would be interesting to explore the implications of the large N factorisation we have described in terms of space-time duals of this form, in particular whether there is some generalization of the connection between multi-particle states in a dual background and the factorization property along the lines of [7]. Double scaled matrix models, which have returned to current interest (see e.g. [101][102][103][104]) should provide interesting settings for the investigation of permutation invariant observables in models with actions invariant under continuous symmetries.