Three-point Functions in $\mathcal{N}=4$ SYM at Finite $N_c$ and Background Independence

We compute non-extremal three-point functions of scalar operators in $\mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three-point functions satisfy the large $N_c$ background independence; correspondence between the string excitations on $AdS_5 \times S^5$ and those in the LLM geometry.


Introduction
Recently we have seen remarkable progress in the computation of the correlation functions of N = 4 super Yang-Mills theory (SYM) in the hope of establishing the AdS/CFT correspondence [1]. There are two complementary approaches to this problem.
The first approach is based on the integrability of N = 4 SYM in the planar limit. The planar three-point functions of single-trace operators are regarded as a pair of hexagons glued together, where each hexagon form-factor is severely constrained by the centrally-extended su(2|2) symmetry [2]. The n-point functions of BPS operators can be studied by hexagonization. The gluing of four hexagons give us the planar four-point functions [3][4][5], and the gluing of 2n−4+4g hexagons should give the g-th non-planar corrections [6][7][8]. Furthermore, certain four-point functions in the large charge limit decompose into a pair of octagons [9,10], which can be resumed [11,12].
The integrability approach tells us how single-trace correlation functions depend on the 't Hooft coupling λ = N c g 2 YM . However, only the non-extremal correlation functions have been studied, because the non-extremality is related to the so-called bridge length (the number of Wick contractions between a pair of operators), which suppresses the complicated wrapping corrections to the asymptotic formula [13][14][15][16][17].
The second approach is based on the finite-group theory. In this approach, one obtains the results valid for any values of N c , though most results are limited to tree-level or a few orders of small λ expansion. In the finite-group approach, extremal correlation functions are often studied, because they are roughly equal to the two-point functions at tree level.
Quite recently the author studied the n-point functions of multi-trace scalar operators at treelevel of N = 4 SYM with U (N c ) gauge group, based on the finite group methods [18]. Those results are written in terms of permutations, meaning that they are valid to any orders of 1/N c expansions, but not at any values of N c because the finite-N c constraints are not taken into consideration. The primary purpose of this paper is to generalize the permutation-based results to finite N c , by taking a Fourier transform of symmetric groups.
Two types of operator bases of N = 4 SYM are well-known, which carry a set of Young diagrams as the operator label, diagonalize tree-level two-point functions at finite N c , generalizing the pioneering work of [19]. The covariant basis (also called BHR basis) introduced in [20,21] respects the global (or flavor) symmetry of the operator. As such, one can construct O(N f ) singlets for general N f [22]. The restricted Schur basis was introduced in a series of papers [23][24][25] and related to multi-matrix models in [26,27]. 1 The restricted Schur basis respects the permutation symmetry of the operator, and suitable for explicit calculation. In other words, one has to specify a state inside the irreducible representation of the global (or flavor) symmetry, like the highest weight state. Here is a brief comparison of the two representation bases [28]: Operator basis Symmetry respected Analogy Covariant Global symmetry Spherical coordinates Restricted Schur Permutation of constituents Cartesian coordinates In this paper, we consider general non-extremal three-point functions of the scalar operators in the restricted Schur basis. There are several important ideas in this computation. The first idea is the Schur-Weyl duality between U (N c ) and S L , which converts powers of N c into the irreducible characters of the symmetric group S L . The second idea is the quiver calculus initiated by [29]. This is a set of diagrammatic rules which enormously simplify the manipulation of representationtheoretical objects. The third idea is the generalized Racah-Wigner tensor. Since the three-point function is non-extremal, we need to compute a non-trivial overlap between the states under different subgroup decompositions of S L . The invariant products we encounter are more general than Wigner's 6j symbols. 2 Let us summarize the main results. Our notation is explained in Appendix A. We are particularly interested in two types of the non-extremal three-point functions (or equivalently non-extremal OPE coefficients). The first type is the super-protected three-point functions [32] in the restricted Schur basis, given by (3.70) Fourier transform of tr L 1 α 1 Z ⊗L 1 tr L 2 α 2Z The second type is the three-point functions of the scalar operators made of three pairs of complex scalars in N = 4 SYM, given by (3.90) Fourier transform of tr The objects G 123 and G 123 are related to the invariant products of the generalized Racah-Wigner tensors.
Mathematically, the branching coefficient of R = ⊕ r,s (r ⊗ s) is the building block of the restricted Schur character and the generalized Racah-Wigner tensor. In the literature, the orthonormal basis of r ⊗ s is called the split basis [33], and the branching coefficients are called fractional parentage coefficients [34], subduction coefficients [35,36] or the split-standard transformation coefficients [33,37,38]. In general, explicit computation of the branching coefficients is a hard problem. See [39][40][41] for the recent results on the branching coefficients, and on the construction of the restricted Schur basis [42].
Likewise, it is difficult to compute G 123 , G 123 explicitly. We conjecture that they can be written by the Littlewood-Richardson coefficients, based on the fact that they satisfy certain sum rules.
From (1.1) and (1.2), it is straightforward to show the large N c background independence in N = 4 SYM [43]. The background independence is a conjectured correspondence between the operators with O(N 0 c ) canonical dimensions and those with O(N 2 c ) canonical dimensions, where the latter is constructed from the former by "attaching" a large number of background boxes. By AdS/CFT, this conjecture implies that the stringy excitations in AdS 5 × S 5 and those in the (concentric circle configuration of) LLM geometry [44].
On the gauge theory side, the large N c background independence has been checked for the case of two-point functions and extremal n-point functions. On the gravity side, some string spectrum of in the SL(2) sector has been studied in [45]. We find that the non-extremal OPE coefficients in the LLM background are essentially given by the rescaling of N c in (1.1), (1.2). Our results provide strong support that the large N c background independence can be found also in the string interactions.

Two-point functions in the representation basis
We review the construction of the restricted Schur basis, and introduce the diagrammatic computation methods called quiver calculus.

Set-up
We consider N = 4 SYM of U (N c ) gauge group at tree-level. This theory has three complex scalars (X, Y, Z), which satisfy the U (N c ) Wick rule, With α ∈ S l+m+n , we define a multi-trace operator in the permutation basis (2. 2) The usual single-trace operators can be expressed in the permutation basis as The correspondence between a multi-trace operator and α ∈ S L is not one-to-one, because α is defined modulo conjugation, which we call the flavor symmetry (or global symmetry). For example, where . . . represents the other permutations generated by the flavor symmetry (2.4).
We define the complex conjugate operator by The two-point function between O (l,m,n) α 1 and O (l,m,n) α 2 at tree-level is given by where C(ω) counts the number of cycles in ω ∈ S l+m+n . We write

Diagonalizing the tree-level two-point
Following [29], we show how to "derive" the representation basis of operators starting from the two-point functions on the permutation basis (2.7). The resulting tree-level two-point functions are diagonal at any N c . The readers familiar with the restricted Schur basis can skip this subsection. The basic formulae are summarized in Appendix A.3.
First, we rewrite the equation (2.7) by using (A.40) as where we used the quiver calculus notation of Appendix B in the second line. We introduce γ = γ 1 • γ 2 • γ 3 ∈ S l ⊗ S m ⊗ S n and the branching coefficients for S l+m+n ↓ (S l ⊗ S m ⊗ S n ) to make use of the identity (A.20). The equation (2.8) becomes We apply the grand orthogonality (B.4) to the matrix elements of γ 1 , γ 2 and γ 3 to obtain where χ R,(r 1 ,r 2 ,r 3 ),(ν + ,ν − ) (α) is the restricted characters defined through branching coefficients, The restricted characters satisfy the orthogonality relations (A.51). It is straightforward to find a linear combination of operators which diagonalizes the two-point function; (2.11) It follows that where we used (A.5).
Recall that O (l,m,n) α in (2.2) becomes half-BPS when l = m = 0, and the restricted character (2.10) reduces to the usual irreducible characters of S n . The two-point function (2.12) becomes which gives the same normalization of half-BPS operators as in [19].

Three-point functions in the representation basis
In [18], tree-level formulae of the n-point functions of general scalar operators in the permutation basis have been derived. We consider three-point functions of scalar operators in the restricted Schur basis below. The three-point functions of N = 4 SYM are related to the OPE coefficients by thanks to the conformal symmetry. By abuse of notation, we write (3.1) as

Set-up
Let us recall the tree-level permutation formula for three-point functions in [18]. That formula has been derived based on the following idea. Consider a non-extremal three-point function of the operators labeled by α i ∈ S L i for i = 1, 2, 3. We expect that the tree-level Wick contractions give the quantity like N . However, we cannot define the multiplication of elements in S L 1 and S L 2 if L 1 = L 2 . This problem can be solved by extending α i toα i ∈ S L for some L , which makes the quantity N Let us explain how this idea works. First, we extend the operator O i by adding identity fields, The permutationα i acts as the identity at the position p at which ΦÂ (i) p = 1. The (edge-type) permutation formula reads We call h ABC a triple Wick contraction.
We will consider two types of three-point functions. The first type is the three-point functions of half-BPS multi-trace operators, The fieldZ belongs to the one-parameter family of operators used in [2,32], The second type is general three-point functions of the scalar multi-trace operator (2.2), where ij is the number of tree-level Wick contractions between O i and O j (called the bridge length), given by and h i is an integer inside the range (3.11)

Partial Fourier transform
We construct the three-point functions in the restricted Schur basis by taking the Fourier transform of C ••• in (3.7) and C XY Z h (3.9). Recall that the usual Fourier transform of the delta function is a constant. In the Fourier transform over a finite group, the Fourier transform of the identity permutation should be a sum over all representations. In other words, if we write then we should sum t i over all possible partitions of L i . In fact, t i is an unphysical parameter, and we can perform a calculation without using t i . Thus we call the procedure (3.12) a partial Fourier transform.
In order to treat C ••• and C XY Z h simultaneously, we extend the multi-trace operator (2.2) as in and define the partial Fourier transform bŷ (3.14) The partial Fourier transform can be rewritten as a linear combination of the complete Fourier transform. To see this, we recall (A.33) and giving us a dummy representation t i to be summed over the partitions of L i . It follows that (3.16) As for C ••• , we introduce the Fourier transform of the half-BPS operators as and defineC As for C XY Z h , we take the Fourier transform of the operators in (3.9) as and defineC We collectively denote the three-point functions of the operators in the representation basis bỹ From (3.5) we get Consider the second line of (3.22). We use the identity (A.40) and (A.9) to obtain We simplify the sum over {α i } in the last line. The character is given by (3.15). We decompose the matrix elements DR I iĴi (α i ) according to the restriction where we introduced the double projector . (3.28) Here we should keep in mind that the restriction to the subgroup of S L is different for each i = 1, 2, 3. We will revisit this issue in Section 3.4.

Now the equation (3.23) is simplified as
where the projector PR →sub I iĴi is given by The three-point function (3.22) becomes where (3.15) is used to sum over t i .

Sum over Wick contractions
We simplify the sum over the Wick contractions, denoted by {U i } ∈ S ⊗3 L in (3.32).

Symmetry of the permutation formula
To begin with, let us review the symmetry in the permutation formula (3.5) for a fixed {U i }, (3.33) SinceC 123 is a linear combination of C 123 , the equation (3.32) should inherit the same symmetry.
which corresponds to the relabeling p → V 0 (p) in (3.33). Second, C 123 ({U i }) is invariant under the permutation of identity fields The redundancy (3.34) and (3.35) are unphysical, which should be canceled by the numerical factors L! and i L i ! in (3.33). The last operation (3.36) is the symmetry of the external operators, and interchanges different Wick contractions.

Fixing redundancy
Let us rewrite the flavor factor p h ABC in (3.33) as Note that the position of each column is unimportant for computing the flavor factor (3.37), We fix the redundancy of V 0 in (3.34) as follows. Let us choose the position of the identity fields for each operator as Here the subscript of 1 is a dummy index, which will disappear after the identification (3.39). The Wick-contraction matrix becomes (1) After the partial gauge fixing (3.40), {U i } permute the non-identity fields only, There is still residual redundancy generated by a combination of V 0 and V i in (3.35), This map does not permute identity fields, but permutes the nonidentity fields sitting in the same column.

Counting inequivalent Wick contractions
We pick up one set of partially gauge-fixed permutations This procedure generates all non-vanishing Wick pairings. To show this, consider two sets of permutations {U • i } and {U • i }, both of which are subject to the partial gauge fixing (3.42) and giving the non-vanishing flavor factor (3.37). Define Since any permutation consists of a product of transpositions, we may assume ( Then, the Wick contractions of {U • i } are written as Since both (3.45) and (3.46) are non-zero, and since Φ = (X, Y, Z) have orthogonal inner products, The sum over (S 1 , S 2 , S 3 ) counts each inequivalent Wick pairing more than once. The multiplicity comes from the residual redundancy (3.43), The number of inequivalent Wick contractions is given by

The OPE coefficients simplified
We collected all non-vanishing Wick contractions by restricting the sum {U i } over the ranges (3.47). The OPE coefficient (3.32) becomes Recall that the projector is equal to the product of branching coefficients, P = B B T as in (3.31). We can simplify the second line by using the identity of branching coefficients (A.21) k across the double branching coefficients B or B T , they annihilate each other; see (3.54).
Let us define a triple-projector product where we used the symbolsP andP to keep in mind that the branching coefficients come from different restrictions of S L . Theñ where we used (3.49).
In the notation of the quiver calculus in Appendix B, we can express the above calculation as From this diagram, we see that IR → sub 123 in (3.52) is also a triple product of the transformation matrices (A.16).

Sum over the triple-projector products
We compute the OPE coefficients by evaluating a sum over the triple-projector products, where the projector is given by (3.31). The main idea is to decompose each projector further into a sum of sub-projectors, so that we can make use of the orthogonality of the sub-projectors on the fully-split space, V F S .
. The Wick-contraction matrix of C ••• after a partial gauge-fixing (3.41) is given by which shows that S i = S L i ⊗ S L i in place of (3.47). We represent (3.56) as in the following figure, Let us choose the fully-split space as On the space V F S , the states decompose as where we used (A.13). We introduce the fully-split branching coefficients by and the corresponding sub-projector by We rewrite the original projectors in (3.31) as a sum over sub-projectors on V F S as By construction, all sub-projectors follow from the same restriction and all sub-representations should be synchronized when evaluating IR → sub 123 in (3.55). The states can also be decomposed as R in addition to (3.60). The consistency of the two decompositions suggests that the multiplicity labels can be rewritten as In (3.63), the representations T i come from the Fourier transform of identity fields 1, and Q i , Q i come from the non-identity fields, Z,Z, Z . Since the OPE coefficient C ••• has the Wick-contraction structure given in (3.57), we should identify the representations We can show (3.67) from another argument. The triple-projector product is equal to the product of generalized Racah-Wigner tensors in Appendix C, which we conjecture as (C. 19), The three-point function (3.53) becomes Here, the Littlewood-Richardson coefficients in G 123 put constraints on the sum over {Q i }. In other words, we should find all {Q i } = {Q i } such that The conditions (3.71) can be summarized as Extremal case. As a check, consider the situation We get and thereforeC This result agrees with the literature [19] including the normalization of the two-point function given in (2.13).

Case ofC XY Z h
Our discussion is quite parallel to Section 3.4.1. Recall thatC XY Z h is a linear combination of C XY Z h given in (3.9). We represent the Wick-contraction matrix bŷ where h i are constrained by (3.11), We choose the fully-split space as and decompose the original projectors (3.31). From (3.76), one finds that the new branch coefficients are needed for (3.79) For example, we rewrite the states for O 1 on the space V F S as and introduce the fully-split branching coefficients by The original projector (3.31) becomes a sum over the sub-projectors P = B B T , and similarlyPR When summing over {t i , t i } we can forget the constraint t i ⊗ t i T i , because the OPE coefficient (3.53) contains sums over {T i }.
All sub-projectors come from the irreducible decompositions ofR under the restriction S L ↓ S F S , R = q ,q ,r ,r ,s ,s g(q ,q ,r ,r ,s ,s ;R) Since the OPE coefficient C XY Z h has the Wick contraction structure of (3.76), we should identify the representations as and replace the multiplicity labels by Again, the trace over the product of sub-projectors is given by the generalized Racah-Wigner tensors (C.27), From the identity of the projectors (A.45), this becomes (3.88) We need to sum over the representations and multiplicity labels. We conjecture that the result is given by (C.38), where M R,r,ν is the slice of the total multiplicity space constrained by (R, r, ν).
The three-point function (3.53) becomes Here {q i , r i , s i } must be consistent with R i in (3.14). This condition is implicitly included in the definition of δ in (C.36). In other words, the OPE coefficients are non-zero only if (q 1 , q 2 , r 1 , r 3 , s 2 , s 3 ) satisfy We find some difference from the case ofC ••• in (3.70). First, we do not have a sum over (q 1 , q 2 , r 1 , r 3 , s 2 , s 3 ). This is becauseC XY Z h has the same structure of the Wick contractions as the extremal correlators for each flavor X, Y, Z. 4 Thus, the first line of (3.91) is trivial. Second, there is no sum over {ν i∓ }, because {ν i∓ } are part of the operator data R i = {R i , (q i , r i , s i ), ν i− , ν i+ }. We should pick up the right combination of multiplicities consistent with R i .
Extremal case. Consider the situation where the operators consist of Z or Z only. This means In particular, we do not need to specify ν i∓ .
The quantity G 123 becomes The three-point function (3.90) becomes which agrees with (3.75) after relabeling.
In Appendix C.3 we consider the restricted Littlewood-Richardson coefficients, which are related to the extremal three-point functions of different type.

Background independence at large N c
We study the tree-level three-point functions in the representation basis, and check the background independence conjectured in [43]. Our proof is based on the conjectured relations for the generalized Racah-Wigner tensor in Appendix C.

The LLM operators
Let us review the argument on the large-N c background independence [43]. They mapped the N = 4 SYM operators with the O(N 0 c ) canonical dimensions to those with the O(N 2 c ) canonical dimensions by attaching a large number of background boxes. We call the latter LLM operators, because they correspond to stringy excitations on the LLM geometry. Recall that the LLM geometries are the half-BPS solutions of IIB supergravity. This implies that the addition of O(N 2 c ) boxes should consist of a single holomorphic scalar like ∼ Z N 2 c .
For simplicity, we consider the operator mixing in the su(2) sector, at one-loop in λ at any N c . We expand the dilatation eigenstates in terms of the restricted Schur basis as We denote the action of the one-loop dilatation on the restricted Schur basis by and define the LLM operator by The operation r → (+ + +r) can be exemplified as We specify a corner of the background Young diagram B, and consider a set of all Young diagrams attached to that corner. This set of states has many interesting properties. First, from the Littlewood-Richardson rule, we find g(r, s; R) g(+ + +r, s; + + +R), (N c 1). (4.5) This allows us to use the same multiplicity labels ν ∓ before and after the + + + operation. Note that the tensor product (+ + +r)⊗s contains representations in which boxes are attached to multiple corners of B. However, the overlap between such states and (+ + +r) is suppressed by 1/N c . Second, the hook length of (+ + +r) factorizes as [43] hook + + +r where η B is the factor which depends only on B, assuming that the small diagram r is put at the C-th corner of B in Figure 1. It follows that Since position of the C-th corner is (i, j) = (1 + D l=C+1 M l , 1 + C k=1 N k ), from (A.5) we get In [43] they found that the operator mixing coefficients satisfy the identity N +R,(+r,s),ν − ,ν + +T,(+t,u),µ − ,µ + N R,(r,s),ν − ,ν + T,(t,u),µ − ,µ + (N c 1) (4.10) showing that

Tree-level OPE coefficients
We revisit two types of OPE coefficients in Section 3. We will show that the OPE coefficients of non-extremal three-point functions in N = 4 SYM are essentially same as those of the LLM operators, after redefinition of N c .

(4.18)
It follows that At large N c , we can simplify this results following our discussion in Section 4.1 as The first line is a numerical prefactor, and the second line agrees with (C XY Z h ) by the redefinition of N c → N c in (4.9).

Conclusion and Outlook
In this paper, we have studied general non-extremal three-point functions of scalar multi-trace operators at tree level valid for any values of N c in gauge theory including N = 4 SYM, by using the representation theory of symmetric groups.
We made full use of various new mathematical techniques. The quiver calculus of [29] gives a collection of diagrammatic method which simplifies various objects in the representation theory. The generalized Racah-Wigner tensor is introduced as an extension of the 6j symbols. We conjectured formulae about the invariant products of the generalized Racah-Wigner tensors, written in terms of the Littlewood-Richardson coefficients.
With these formulae, we provide strong evidence on the large N c background independence, a correspondence between small (O(N 0 c )) and huge (O(N 2 c ) operators of N = 4 SYM. The background independence has been checked for two-point functions as well as extremal three-point functions. Our argument demonstrates that it extends to non-extremal three-point functions. These results will clarify the properties of stringy excitations on the LLM backgrounds, particularly how they differ from the usual strings on AdS 5 × S 5 .
Let us comment on some important future directions.
The first direction is to find a connection with the integrability results of the planar N = 4 SYM. Clearly, the operators in the representation basis are not the eigenstates of the dilatation operator of N = 4 SYM. One should think of the representation basis as a tool for the finite N c computation. The two-point functions of single-trace operators in the su(2) sector have been computed in this way [27,46], generalizing the old results of the complex matrix model [47,48]. A particularly interesting question is to determine the so-called octagon frame, namely the tree-level part of the "simplest" four-point functions of N = 4 SYM in the large charge limit [11]. The finite group methods developed in this paper can be used for the exact finite -N c computation, because it is a generalization of the character expansion methods familiar in the matrix models [49][50][51], and The second direction is to refine our computation. The conjectured formula for the invariant products of generalized Racah-Wigner tensor should be proven. The computation of the n-point functions in the representation basis is also important. It is interesting to ask whether one can bootstrap four-point functions out of two-and three-point data.
The third direction is to investigate a possible relation between quiver calculus and knot theory. The 6j symbol of the unitary group has been extensively studied in the context of knot theory and integrable systems [52]. Since the 6j symbols of symmetrical groups are related to those of unitary groups, the quiver calculus could give a new insight into the study of knot polynomials. For example, some non-trivial conjectures about the 6j symbols have been made [53][54][55], though most of them discuss the multiplicity-free cases only. Since the new invariants G 123 and G 123 discussed in this paper are closely related to the multiplicity structure, studying similar quantity in the case of unitary groups is a fascinating problem.

RS thanks Robert de Mello Koch and Sanjaye Ramgoolam for their comments on the manuscript, and is obliged to Korea Institute for Advanced Study where this research has been initiated.
A Survey of finite-group representation theory We explain our notation and formulae used in the main text, while providing a brief survey of the representation theory of finite groups. Our notation is similar to the one used in [22]. For more details on finite groups, see textbooks like [71,72].

A.1 Basic notation
The symmetric group permuting L elements is denoted by S L . We denote the conjugacy class of S L by A permutation cycle is denoted by (12 . . . L) ∈ Z L . Any element of S L consists of permutation cycles. The number of length-k cycles in σ ∈ S L is denoted by Cyc k (σ). The number of cycles in σ is so that C(id) = C((1)(2) . . . (L)) = L.
A partition of L, or equivalently a Young diagram with L boxes, is denoted by R L. Define where d R is the dimension of R as the representation of S L , and Dim N (R) is the dimension of R as the representation of U (N ). 5 For example, hook R and Wt N (R) of the Young tableau R = are given by We assume that all representations are real and orthogonal. 6 Denote the I-th component of the irreducible representation R of S L by R I , with I = 1, 2, . . . , d R . Introduce the dual basis by Let D R IJ (σ) be the representation matrix of σ ∈ S m+n of the representation R L, The character of the representation R for the group element σ is denoted by 7 By restricting σ ∈ S L = S m+n to S m ⊗ S n , we obtain the irreducible decomposition 8 R = where g(r, s; R) is the Littlewood-Richardson coefficient. It counts the number of r ⊗ s appearing in the irreducible decomposition of R. The subscript ν is called the multiplicity label. With an appropriate change of basis, we can transform the representation matrix into a block-diagonal form, such that it matches (A.10). By definition of the irreducible decomposition, there are no off-blockdiagonal elements including the multiplicity labels. For general σ ∈ S m+n , the matrix (A.11) has off-block-diagonal elements. 9 Let r,s i,j ν be an orthonormal basis of r ⊗ s at the ν-th multiplicity, satisfying for ν k = 1, 2, . . . , g(r k , s k ; R). The rotation matrix is called the branching coefficients, defined by

A.2 Branching coefficients
We find from (A.11) that the branching coefficients satisfy the completeness relations = δ I,J (A.14) In (A.15), we assume that two product representations r 1 ⊗ r 2 and s 1 ⊗ s 2 descend from the same restriction S m+n ↓ (S m ⊗ S n ). If they descend from different restrictions, then the two branching coefficients B andB are unrelated, and we obtain another orthogonal matrix For example, given two irreducible decompositions any pairs r 1 ⊗ r 2 and s 1 ⊗ s 2 from different restrictions can have non-vanishing overlap, e.g.
Sometimes we take the coordinates explicitly in order to distinguish S m+n ↓ (S m ⊗ S n ) and S m+n ↓ (S n ⊗ S m ). For example, the following two restrictions to (A.20) and summing over J, we find .

(A.21)
Again, by multiplying (B T ) to (A.21) and summing over J, we find In the RHS, the matrix elements of γ 1 • γ 2 in the split basis are independent of the multiplicity labels µ, ν. This can be understood also from the construction of the Young-Yamanouchi basis.
The branching coefficients (A.13) for general restriction S L ↓ (S m 1 ⊗ S m 2 ⊗ · · · ⊗ S m ) are given by

A.3 Restricted Schur basis
Consider the restriction S M ↓ (S m 1 ⊗ S m 2 ⊗ S m 3 ) with M = m 1 + m 2 + m 3 , which corresponds to the multi-trace operators with three complex scalars in (2.2).
Define the restricted Schur characters by using the branching coefficients [29], Define the operator in the restricted Schur basis by (A.25) The inverse transformation from the restricted Schur basis to the permutation basis is which can be checked by the row orthogonality of the restricted characters (A.51), As discussed in Section 2.2, the tree-level two-point function is hook R hook r 1 hook r 2 hook r 3 δ RS δ r 1 s 1 δ r 2 s 2 δ r 3 s 3 δ ν + µ + δ ν − µ − . (A.28)

A.4 Formulae
The formulae for the irreducible characters and the restricted characters will be summarized below. For simplicity, we mostly consider the restriction S m+n ↓ (S m ⊗S n ). Generalization to S M ↓ (⊗ k S m k ) is straightforward.
Character Orthogonality. Let R, S be the irreducible representations of S L . The representation matrices satisfy the grand orthogonality relation By taking the trace, we obtain the row (or first) orthogonality relation of irreducible characters, The irreducible characters also satisfy the column (or second) orthogonality relation, where |C σ | is the number of elements in a given conjugacy class (A.1). This relation follows from the fact that any class function can be expanded by irreducible characters As a corollary, the δ-function can be written as Multiplicity label. There are several ways to understand Littlewood-Richardson coefficients.
(A. 46) It follows that The restricted projector is useful for fixing the normalization. These formulae as well as the following identities can be proven by using the quiver calculus in Appendix B.
Restricted Character Orthogonality. The restricted characters (A.24) satisfy the identities where the last relation is consistent with (A.22). The row and column orthogonality relations (A.31) are generalized as One can generalize the grand orthogonality relation (A.29) with the branching coefficients in two ways. First, let R and S be the irreducible representations of S m+n . A sum over S m+n gives which reduces to (A.51) by taking the trace over r 1 ⊗ r 2 = s 1 ⊗ s 2 . Second, let (r 1 , r 2 ) and (s 1 , s 2 ) be the irreducible representations of S m ⊗ S n . A sum over S m ⊗ S n gives where we used (A.22)

B Quiver calculus
Let us introduce a graphical notation of various representation-theoretical objects following [29]. We denote the indices of R L = (m + n) by a double line, and those of r 1 m or r 2 n by a single line. We use different lines to distinguish two set of representations {R, (r 1 , r 2 )} and {S, (s 1 , s 2 )}.
The matrix representation of a permutation group element is represented by The grand orthogonality relation (A.29) is or equivalently The branching coefficients (A.13) are represented as The character and the restricted characters are We can show the row orthogonality of the restricted character as To show the column orthogonality, we insert the resolution of identity on the irreducible representation R by (A.29), We obtain where we used (A.33). Note that Similarly, we can derive the column orthogonality for the restricted characters (A.52). By using (B.16) In the last line, we cannot use (B.14), because γ ∈ S m ⊗ S n S m+n .
We can show the restricted grand orthogonality (A.53) by Restricted projector. The restricted projector (A.43) can be represented as which is an element of C[S m+n ] and not a number. Its matrix elements are given by the branching coefficients (A.46), which can be shown by The identity (A.45) follows from the calculation

C Generalized Racah-Wigner tensor
The associativity of triple tensor-product representations gives rise to the 6j symbols, which is also called Wigner's 6j invariants [73], Racah W -coefficients [74] or recoupling coefficients [75], The problem of computing 6j symbol is called the Racah-Wigner calculus.
We construct a slightly general object from the branching coefficients. The generalized 6j symbol is covariant under the action of symmetric groups, and contains four multiplicity labels.

C.1 Case ofC •••
Consider two ways of the double restriction with L = L 1 + L 2 + L 3 , which corresponds to the calculation ofC ••• in Section 3.4.1. They induce the irreducible decompositionŝ The corresponding branching coefficients are The multiplicity labels (µ, ρ) and (µ , ρ ) run over the spaces which are subsets of the total multiplicity space induced by the irreducible decomposition η ∈ M tot , |M tot | = g(q 1 , q 2 , q 3 ;R). (C.6) From the identity (A.39), we obtain the following relation between the branching coefficients in (C.4) and (C.6), q 1q2q3 abcη where the RHS depends on R 12 through the multiplicity space of (µ, ρ) in (C.5).
(C. 17) We conjecture that both sides are equal, and continue the discussion below. Similarly, we find A solution to these equations is In view of (C.14), our conjecture is summarized as (C.20)
These products are depicted as The identity of the projectors (A.45) suggests that (C.32) By summing {ξ ∓ , ξ ∓ , ξ ∓ } over the ranges {M R ∓ ,Q ∓ ,ν ∓ , M R ∓ ,Q ∓ ,ν ∓ , M R ∓ ,Q ∓ ,ν ∓ }, we discover the overlap As a solution to the sum rules, we conjecture that where δ νν should be understood as the intersection inside M tot (C. 36) It follows that

C.3 Restricted Littlewood-Richardson coefficients
Let us compute the restricted Littlewood-Richardson coefficients in [27] in our method. We will find the perfect agreement. However, they considered multiplicity-free cases only. Thus, this agreement does not provide non-trivial checks of our conjectured formula.
We define the restricted Littlewood-Richardson coefficients by L i = m i + n i , R i = {R i , (r i , s i ), (ν i− , ν i+ )} .