The MacMahon R-matrix

We introduce an R-matrix acting on the tensor product of MacMahon representations of Ding-Iohara-Miki (DIM) algebra Uq,tgl^^1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {U}_{q,t}\left({\widehat{\widehat{\mathfrak{gl}}}}_1\right) $$\end{document}. This R-matrix acts on pairs of 3d Young diagrams and retains the nice symmetry of the DIM algebra under the permutation of three deformation parameters q, t−1 and tq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{t}{q} $$\end{document}. We construct the intertwining operator for a tensor product of the horizontal Fock representation and the vertical MacMahon representation and show that the intertwiners are permuted using the MacMahon R-matrix.


Introduction
Ding-Iohara-Miki (DIM) algebra [1,2], also denoted by U q,d ( gl 1 ), or U q,t ( gl 1 ) is a tdeformation of the algebra of volume-preserving diffeomorphisms of the quantum torus [3,4], which can also be described as qW 1+∞ -algebra, or a deformation of the double loop algebra of gl 1 = C × [5]. The DIM algebra is an extremely elegant object. It has three deformation parameters q 1 = q = dq −1 (coming from the quantization of the torus), q 2 = t −1 = q −1 d −1 (the t-deformation) and q 3 = t q = 1 q 1 q 2 = q 2 , which are treated on JHEP04(2019)097 equal footing. In fact, the algebra is invariant with respect to the permutations of q i . It also enjoys an SL(2, Z) symmetry, 1 corresponding to the mapping class group of the quantum torus. There are two central charges in the algebra, which form a two-dimensional representation of the SL(2, Z) automorphism group. A natural 2 representation of the DIM algebra seems to be the MacMahon module M(K; v), i.e. a linear space spanned by plane partitions, or 3d Young diagrams. Here is an example of a 3d Young diagram: = [ [2,1], [1]]. (1.1) We denote the 3d diagrams as sequences of ordinary Young diagrams lying at the successive horizontal layers, as seen on the r.h.s. of eq. (1.1). Metaphorically, one can think of an additional dimension in the diagrams as related to an additional loop in the definition of the algebra (compared, e.g., to the affine or Virasoro algebras). The MacMahon representation has the central charge vector (1, K) (with K arbitrary). Permutations of the three deformation parameters q i act on this representation by permutations of the axes of the 3d Young diagram, i.e. by 3d "transpositions".
There are other representations of the DIM algebra, which are in fact reductions of the MacMahon one. If one puts the central charge K to a particular quantized value K = q i , then the MacMahon representation becomes reducible. After factoring out the invariant subspace, one gets the space of ordinary Young diagrams. The resulting representation is called the Fock (or free field) representation F (v). More generally, when K = q a 1 q b 2 q c 3 , a, b, c ∈ Z, the MacMahon representation reduces to a representation spanned by plane partitions with a "pit" [10,11]. Further reduction is provided by the vector representation, in which the central charges are trivial and the vectors can be thought of as columns of boxes, i.e. 1d Young diagrams.
The DIM algebra is a (quasi triangular) Hopf algebra and has a universal R-matrix. In [12,13], we have explicitly constructed the R-matrix acting on the product of Fock representations and elucidated its role in refined topological strings and the Nekrasov partition function. 3 In the present work, we follow the same program and construct the DIM R-matrix for the MacMahon representations.
For the Fock spaces, there is a well-known construction [15] of intertwining operators, which turn out to be the refined topological vertices C λµν (q, t) of topological string theory [16][17][18][19][20][21][22][23]. The labels λ, µ and ν in the vertex are 2d Young diagrams denoting the vectors in the three Fock spaces being intertwined. Here we are going to generalize the intertwiner construction so that one of the labels in the vertex is the 3d Young diagram. This indicates the existence of a generalization of refined topological string theory, in which the 3d Young diagrams live on the edges along the preferred direction of the toric diagram.

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The toy algebra. Instead of the DIM algebra, we take its t = 1 limit: qW 1+∞ algebra, which is a Lie algebra spanned by the generators W m,n satisfying the relations [W m 1 ,n 1 , W m 2 ,n 2 ] = (q n 1 m 2 − q n 2 m 1 )W m 1 +m 2 ,n 1 +n 1 + (c 1 n 1 + c 2 m 1 )q n 1 m 1 2 δ m 1 +m 2 ,0 δ n 1 +n 2 ,0 , (1.2) where c 1,2 are the central charges. There are two grading operators d 1 and d 2 counting m and n in W m,n . The relations are manifestly SL(2, Z)-invariant, if we assume that the labels (m, n) and the central charges (c 1 , c 2 ) transform as 2d vectors under SL(2, Z)transformations.
The toy representation. We further simplify our toy model by putting c 1,2 = 0. Then the algebra (1.2) can be represented by difference operators acting on functions of one variable z: ρ(W m,n ) = z m q nz∂z .
We call representation ρ the vector representation. This is nothing but the evaluation representation of the algebra (1.2). As usual, there are two viewpoints on the evaluation representation. In the first convention, the representation is seen as a finite-dimensional (in our case one-dimensional) space of functions f (z), while, within the second approach, the representation is an infinite-dimensional space of vectors, in which the parameter z is not a variable, but is fixed. In the case under discussion, one can also use the equivalence of two conventions and say that the states in the vector representation, instead of being functions of z, are now labelled by an integer and a complex parameter, |z, i , so that ρ(W m,n )|z, = z m |z, + n . (1.4) The parameter z is fixed in this definition and is called the spectral parameter of the vector representation. Evidently, this is just a rewriting of eq. (1.3) with |z, corresponding to f (zq ).
The toy intertwiner. The toy algebra (1.2) is a Hopf algebra with the standard Liealgebraic coproduct ∆(W m,n ) = W m,n ⊗ 1 + 1 ⊗ W m,n . (1.5) One can then build tensor products of representations. In particular, one can take the tensor product of the Fock representation of (1.2), which has c 1 = log q 3 K 1 = 1, c 2 = log q 3 K 2 = 0, and the vector representation. There is an intertwining operator Ψ, which acts from this tensor product into a single Fock representation. It is just the standard (q-deformed) vertex operator The intertwiners commute up to contact terms, which is the reflection of the fact that the coproduct (1.5) is symmetric, and thus the R-matrix of the algebra (1.2) is trivial.

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A small generalization and the toy R-matrix. Leaving the rigorous derivation for section 4, we give here the vector intertwining operator for t = q (see also [28]): The commutation of intertwiners (1.7) yields a nontrivial (though diagonal) R-matrix: This R-matrix, though may seem simple, actually encodes in itself the information about all the representations of the DIM algebra. This is because all the representations can be built from the vector representation by taking appropriate tensor products so that the intertwiners also become products of the basic intertwiner (1.7). This will be precisely our strategy in the main part of the paper.
2 Representations of DIM algebra 2.1 Definition of the quantum toroidal algebra U q,d ( gl 1 ) The quantum toroidal algebra U q,d ( gl 1 ) has two deformation parameters q, d. We introduce the structure function where our convention is In this paper, we assume that q i are generic, that is, q a 1 q b 2 q c 3 = 1 for integer a, b, c if and only if a = b = c. We also use The generators of U q,d ( gl 1 ) are E k , F k , K ±1 , H r (k ∈ Z, r ∈ Z \ {0}) and the central element C. It is convenient to introduce the generating functions (currents) H ±r z ∓r . (2.5)

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Note that K ± (z) is expanded in negative (positive) powers of z. With these currents, U q,d ( gl 1 ) is defined by the following relations: 4 (2.12) The coefficientg in (2.10) affects only the relative normalization of currents E(z), F (z) and K ± (z). In this paper, we chooseg There are essentially 5 two central elements C and K − = (K + ) −1 , and we define that the representation has level (γ 1 , γ 2 ) if C = γ 1 and K − = γ 2 . We call representations with C = 1 and C = q vertical and horizontal representation, respectively. 6 The commutation relation of modes H r are read from (2.7), where we define q-integer by Note that in the vertical representations H r are mutually commuting. Actually the vertex operator representation (see section 2.3), which is the only horizontal representation to be considered in this paper, satisfies stronger relations: They also satisfy the Serre relations, which we do not use in this paper, [2,29]. 5 We impose K − = (K + ) −1 . 6 They correspond to the two choices of the Borel subalgebra, "vertical" and "horizontal" are related by the spectral duality automorphism S [2, 30-38] (a proof is given in Lemma A.5 of [27]), which is one of the generators of the SL(2, Z) automorphism group of the DIM algebra.

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To define the tensor product of two representations, we need a coproduct of U q,d ( gl 1 ). We use the following coproduct in this paper: where C 1 = C ⊗ 1 and C 2 = 1 ⊗ C.
We also have two grading operators d 1 and d 2 such that (2.31)

Vertical representations
Since C = 1 for the vertical representations, the modes H r (r ∈ Z\{0}) commute with each other due to the relation (2.7). Then we can find a basis which simultaneously diagonalizes H r . In [11,26], three kinds of vertical representations: the vector, Fock and MacMahon representations have been defined. In these representations, the basis on which H r acts diagonally is roughly speaking labeled by 1D, 2D and 3D Young diagrams, respectively. Accordingly, we can define the Fock representation as an irreducible subrepresentation of an appropriate infinite tensor product of the vector representations [26]. In a similar manner, we can construct the MacMahon representation from the Fock representations [11]. The second level of the vector representation is γ 2 = 1. However, the regularization procedure required in the above procedure makes γ 2 non-trivial. Consequently, the Fock representation has a quantized level γ 2 = q. Moreover, quite interestingly the MacMahon representation allows continuous level γ 2 = K. We can find a natural regularization for the Fock representation, while that for the MacMahon representation somehow looks ambiguous and leads to an arbitrary value K, which we can interpret formally as a limit of q N (N → ∞).

Vector representation
The vector representation V (v) consists of the basis [v] i labeled by i ∈ Z and depends on the spectral parameter v. The action of the algebra is defined as where the generating function of eigenvalues of K ± (z) is expressed bỹ The multiplication factors E and F are determined from the choice ofg: and, in this paper, we choose Sinceψ(0) = 1, the vector representation has the trivial level (1,1). It is also convenient to introduce a more fundamental function ψ(z) such that (2.39)

Fock representation
At first following [26], we outline the idea to obtain the Fock representation (ρ, F (v)). One can construct a natural tensor product of the vector representations by making use of the coproduct of U q,d ( gl 1 ). We want to define an infinite tensor product of the vector representations and find an irreducible subrepresentation, whose basis is composed of partitions, that is, the Fock representation. Let us consider the following tensor product with q 2 -shifted spectral parameters: Since C = 1 for the vector representations, the coproduct (2.27) and (2.28) give

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Similarly, the coproduct (2.25) and (2.26) implies The tensor product representation is defined by ρ N (X(z)) : does not form an inductive system because actions of ρ N and ρ N +1 on |λ) (λ ∈ Z N ) are different. For this reason, one cannot take a limit N → ∞ naively. In order to settle this problem, we modify the action of (ρ N , ) so as to form an inductive system of U q,d ( gl 1 )-modules. Then one can take the inductive limit ρ = lim N →∞ρN to find the Fock representation (ρ, F (v)) as an irreducible subrepresentation of ρ: where λ = (λ 1 , . . . , λ (λ) ) ∈ P : set of partitions and λ n = 0 (n > (λ)).
In the following, we describe the idea of modification in detail. Actually, we have to modify only ρ N (K ± (z)) and ρ N (F (z)) keeping ρ N (E(z)) intact. Let us modify the action of ρ N toρ N so as to obtain the condition, for X = K ± , E or F , Then we can define the action of ρ(X(z)) on |λ) (λ ∈ Z N −1 ) asρ N (X(z)). Therefore, we should search for a modified action which satisfies the condition (2.45). At first, let us see that the actionρ N (E(z)) can be the same as ρ N (E(z)). This action satisfies the condition (2.45) due to the vanishing property, Further, let us focus on the action of K ± (z). This time the actions of ρ N and ρ N +M differ. We denote one gets a recursion relation for the modification factor β +

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This is equivalent to which determines β ± N up to an appropriate initial condition. Naturally, the initial condition should be related to the regularization problem of the vacuum of the Fock representation. To see this, let us look at the unmodified representation ρ N on the "vacuum state |∅)", This expression makes no sense in the limit of N → ∞, but we can formally regularize it by specifying the ordering of infinite products, Therefore, in the modified representationρ N , one expects the vacuum expectation value to be Thus, the problematic factor (2.51) has been replaced by the factor q −1 by the regularization. Now (2.53) leads to the initial condition Hence, the above prescription for the regularization gives As concerns the action of F (z), we also need some modification factor. This factor should be precisely the same as β + N (v/z) for the sake of (2.10). In fact, it satisfies the condition (2.45) due to the vanishing property,

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Now we can write down the action of the infinite tensor product representation (ρ, F (v)) explicitly (see (2.41)-(2.43)), As already mentioned, we can find the invariant subspace F (v) which consists of only partitions by investigating the positions of zeros appearing in the action of the creation operator E(z) and the annihilation operator F (z) [26]. This irreducible subrepresentation generated by the empty Young diagram ∅ is called Fock representation. The Fock representation is the highest weight representation with the empty Young diagram ∅ being the highest weight state. The generating function of eigenvalues of the vacuum is (∅|K + (z)|∅) = ψ(q 3 v/z) −1 . Note that, similarly to the vector representation, the finite tensor representations have trivial level (1, 1), however, now the Fock representation has the nontrivial level (1, q) due to the chosen regularization (2.53).
Thanks to the relation one can rewrite the above formula in a symmetric way, for example,

MacMahon representation
We want to define the MacMahon representation as an appropriate limit of (π N , M N (v)) similarly to the definition of the Fock representation [11], (2.64) The notation of the plane partition (3d partition) Λ is as follows and we also use the notations We interpret as |Λ (k) ) ∈ F (q k−1 3 v) unless otherwise mentioned. Let us now focus on the action of K ± (z) and the regularization problem. We denote we get a recursion relation for the modification factor γ + N as Hence, we can determines γ ± N up to an appropriate initial condition. As concerns the initial condition, let us look at the unmodified representation π N on the vacuum state This expression makes no sense in the limit of N → ∞, but we cannot use the same strategy as in the previous subsection due to the presence of the monomial factor q −N . Here we have to regularize it in another way: we formally substitute q N by an arbitrary parameter K 1/2 . Then, in the modified representationπ N , we have

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which leads to the initial condition Hence, our regularization gives Now we can write down the action of the MacMahon representation (π, M(K; v)), where we have introduced the coordinates x ijk = q j−1 1 q i−1 2 q k−1 3 = q j 1 q i 2 q k 3 and the generating function of eigenvalues of the vacuum: We note that j = Λ (k) i is understood at the r.h.s. of (2.76) and (2.77). The action of K ± (z) has a manifestly symmetric expression with respect to the permutation of q 1,2,3 but E(z), F (z) do not. In [11], it was proved that the actions (2.75)-(2.77) indeed keep the subspace spanned by plane partitions. This is assured by the q 3 -shift of spectral parameters among adjacent Fock representations in the tensor product. One can check that the factor i , which means that the action of E(z) cannot add the box at (i, Λ i , k). Thus the invariant subspace M(K; v) is generated by acting E(z) on the empty plane partition, which is the highest weight state, while the action of F (z) keeps M(K; v). In fact, there is a formula called shell formula [11,39], which expresses the r.h.s. of (2.76) and (2.77) as JHEP04(2019)097 a finite sum over the concave and the convex corners of Λ. Note that π N has level (1, q N ) as an N -fold Fock representation, however, now the MacMahon representation has level (1, K 1/2 ) with a continuous parameter K due to the regularization (2.72).

Horizontal vertex operator representation
When C = q, the Heisenberg part satisfies and we define the fundamental vertex operator as follows, The fundamental OPE by the normal ordering is where the scattering factor s(z) is and satisfies the following formulas as a rational function, We can define the vertex operator representation of level (q, 1) by making use of the fundamental vertex operator as follows: (2.85) where the vacuum state |0 of the Fock space for the horizontal representation is defined by the annihilation condition The shift of the argument in K ± (z) is conventional. Furthermore, for any γ 2 ∈ C × , we can employ a more general level (q, γ 2 ) representation with zero modes e(z), f (z), k ± (z),

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where the constant part of k ± (z) is fixed by the second level γ 2 . We denote the horizontal representation with these zero modes by H = H(k ± (z), e(z), f (z)). The zero modes must satisfy appropriate relations. In fact, we have from (2.10), so e(z) can be determined from f (z) and k ± (z) uniquely Furthermore, we make an ansatz that k ± (z) is independent of z: where u is the spectral parameter of the representation. With the notation introduced above, we can express this horizontal representation as F (q,q N ) It is also useful to introduce the dual vertex operatorṼ ± (z) that satisfies It is expressed explicitly as

DIM R-matrix for MacMahon representations
In this section, we calculate the universal DIM R-matrix in the basis of generalized triple Macdonald polynomials. It will be diagonal and depend on the ratio of spectral parameters x = u v , two central charges K 1 , K 2 and a pair of plane partitions Π, Λ. We use the formula for the diagonal (on the vertical representations) part of the universal R-matrix from [27]: where K is the second component of the central charge vector, d 1 is the first grading operator and h n are modes of the K ± (z) currents defined as follows: For the vertical MacMahon representation, we have and therefore The R-matrix in two MacMahon representations is given by (we divide by the vacuum matrix element to get rid of the overall scalar factor, which can be evaluated separately) . The result is a rational function, e.g.
It reduces to the R-matrix for the Fock representations in the limit of K 1,2 → q 3 = 1 q 1 q 2 . Notice that the R-matrix obeys a simple inversion identity which does not allow one to use such an R-matrix for generating new knot invariants [13, section 12]. It also respects the symmetry of the DIM algebra under arbitrary permutation of parameters q i , which needs to be accompanied by the corresponding transposition of the plane partitions Π and Λ, e.g. exchange of q i and q j corresponds to the transposition in the (i, j)-plane.
In what follows, we build intertwining operators of the Fock representations with the MacMahon ones and verify that the same R-matrix determines their commutation relations.

Intertwining operator for MacMahon representation
The intertwiner of the quantum toroidal algebra U q,d ( gl 1 ) is the vertex operator that intertwines the tensor product of vertical and horizontal representations with an appropriate horizontal representation. Graphically, it is represented by a trivalent vertex with two "horizontal" edges and a single vertical edge. As we explained earlier, the center of U q,d ( gl 1 ) is two dimensional, and one defines two levels (γ 1 , γ 2 ) for each representation. In the following, we fix the horizontal representation as the vertex operator representations of free deformed bosons. It has the unit first level γ 1 = q. On the other hand, we call representations with γ 1 = 1 vertical representation, 7 and there are various choices for the vertical representations, as we saw in the previous section. Note that the condition γ 1 = 1 is kept intact under taking the tensor product of two vertical representations.
In  Since C 1 = 1 and C 2 = q for the vertical and the horizontal representations, the definition of the coproduct (2.25)-(2.28) implies the following intertwining relations for the 7 In the case of affine algebra, the evaluation representation is an example of the vertical representation.

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α-components: Note that, in the above formulas, operators at the right side of Ψ α act on H, while operators on the left side of Ψ α act on H . The matrix elements (β|X|α) are computed in V. Our tasks are to construct the component of the intertwiner in the vertex operator formalism (section 2.3) and to specify admissible horizontal representations. Note that we can typically specify the relative conditions between H and H but there seemingly remains some freedom for each of them. At first, we will construct the vector intertwiner, which is essentially the same as the one given by [28] before. Then, as a composition of the vector intertwiner with an appropriate ordering, we will obtain the Fock intertwiner firstly given by [15]. Finally, using the same strategy we will construct the MacMahon intertwiner from the Fock intertwiner. The N -fold Fock intertwiner was constructed in [40], but the spectral parameters of Fock representations were independent in their construction. Our MacMahon intertwiner corresponds to an appropriate N → ∞ limit of the intertwiner in [40] when the spectral parameters are correlated so that the infinite tensor product leads to the MacMahon representation.
In the following, we use the notation x(z) for the zero modes of the vertex operator representation H, while x (z) for those of H , where x = e, f , k ± . We also denote the second level of H, H by γ, γ unless otherwise mentioned.

Definition of the vector intertwiner
Let us consider the intertwining operator for the vertical vector representation V (v). We define the n-component of the vector intertwiner by where {[v] n−1 } n∈Z is the basis of V (v). Explicit intertwining relations for I n (v) are In the next subsection, we find that there are consistency conditions between the source H and the target H , namely γ = γ on the second level and e (z) = q −1 2 e(z), f (z) = q 2 f (z) on the zero modes.

Construction of the vector intertwiner
We define the operator I n (v) between two horizontal representations as where z n = z n (v) is a stack of zero modes (4.14) Note thatĨ n (v) is a lift of the function (2.36) to the vertex operator, and we can formally interpretĨ n (v) as an infinite product form Using the fundamental OPE relations (4.20) Actually, we have defined I 0 (v) so that it satisfies the relations (4.17), (4.18), which imply γ = γ on the second level. The relative relation for the zero modes is determined from the intertwining relations for F (z) and E(z). To see it, let us check the intertwining relation for the 0-component for F (z). To obtain the relation (4.11), we find that the additional factor q 2 of F (z) on the left is necessary, compared to F (z) on the right, namely

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where we have used the relation (2.91) and This is the reason why the vector intertwiner should shift the horizontal zero mode by q −1 2 . In the same way, one can check The intertwining relation for general components can be also checked easily by making use of the relation among the functions G(z),ψ(z) and s(z).
If we choose the horizontal representation H = F (q,q N ) u used in [15], the zero mode sector is defined by (2.94) and H = F (q,q N ) q −1 2 u . Then, the zero mode stack z n = z n (N ; u|v) is Essentially the same vector intertwiner was given in the Appendix of [28], where they computed the generating function of the components of the intertwiner (4.7). By the simple n-dependence (4.12) of the components, it is easy to see that the intertwiner in [28] agrees with ours.

Definition of the Fock intertwiner
Let us consider the intertwining operator for the vertical Fock representation F (v). With the basis {|λ)} of F (v), explicit intertwining relations for the λ-component Φ λ (v) are where matrix elements can be read from (2.57)-(2.60). In the next subsection, we obtain as consistency conditions the relative shift γ = qγ of the second level and e (z) = (−qv/z)e(z), f (z) = (−qv/z) −1 f (z) on the relative shift of the zero modes.

Construction of the Fock intertwiner
We construct the Fock intertwiner in a parallel way to the construction of the Fock representation (section 2.2.2) in contrast to a rather direct way of [15]. In our approach, taking the tensor product of the vertical representations is realized by merely composing the corresponding vector intertwiners. And the modification factor β ± n (v/z) is lifted to the corresponding vertex operator B n (v).
We begin with specifying the modification operator B n (v). This is realized by making use of the dual vertex operator (2.97) as and satisfies the following relations Note that we can formally interpret B n (v) as an infinite product form Now we can define the operator Φ λ (v) between two horizontal representations as the following composition 8 of vector intertwiners with the modification operator B n (v) is defined by the normal ordering 9Ĩ [n] (4.38) 8 One should be careful of the ordering of constituent vector intertwiners. 9 We have defined as λn = 0 for n > (λ).

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Finally, z λ = z λ (v) is a stack of zero modes (see the discussion of the intertwiner zero modes in appendix C) Note that the definition (4.37) is independent of n > (λ) thanks to the relation :Ĩ [n] which can be also understood from the formal infinite product form of them. The consistency conditions between the horizontal representations in Φ λ (v) : H → H can be read as Indeed, these zero modes of H satisfy the relations (2.91). We can confirm that the operator Φ λ (v) agrees with the AFS intertwiner [15], and the factor G λ , which is defined independently of n > (λ) bỹ plays the same role as c λ in [15]. An explicit form of G λ is where n(λ) = (λ) j=1 (j − 1)λ j and x i,j = q j−1 1 q i−1 2 .

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One can check the Fock intertwining relations by making use of the vector intertwining relation. For example, to check the intertwining relation with E(z), one can compute as follows: where we have introduced the "weighted" commutator by Hence, we obtain As concerns the intertwining relation with F (z), one can compute as follows: (4.50) Note that this time we do not have to use the "weighted" commutator. Again, the last term vanishes and, by substituting the intertwining relations for the vector intertwiner, we where the matrix elements can be read from (2.75)-(2.77). In the next subsection, we specify the constraint on the relations among the zero modes.

Construction of the MacMahon intertwiner
We construct the MacMahon intertwiner in the same way as the construction of the Fock intertwiner in section 4.2. We begin with specifying the vertex operator Γ n (K; v) that produces the modification factor γ ± n (z/v) given by (2.74) in the OPE relation. This operator is realized by

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and satisfies the following fundamental relations (4.60) Note that we can formally interpret Γ n (K; v) as an infinite product form as Then we find the following relations: which mean that the operator Γ n (K; v) produces the modification factor γ ± n (v/z) up to a monomial factor. This discrepancy corresponds to a shift of the second level: Furthermore, the normal orderings with ξ and η are evaluated to give In order to obtain the relation F (z)Γ n (K; v) − Γ n (K; v)F (z) = 0, we need the relative shift where the theta function θ q 3 (z) is defined by (B.1). Note that this in turn determines the relative shift for e(z). As a result, we have

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resulting partition functions and corresponding constraint algebras [8,[41][42][43][44]. It is also very interesting to understand if there is a corresponding Type IIB string construction. Since a pair of central charges of the DIM representation corresponds to the (p, q)-charges of the 5-brane in Type IIB, it is not immediately clear what would the MacMahon representation with an arbitrary (non-integer) central charge correspond to.
Using the technique of [45,46], one can derive the (q, t)-KZ equations for the DIM intertwiners containing not only Fock representations, but the MacMahon ones too. However, in this approach, the MacMahon representations are restricted to lie only on the "vertical" legs. It would be interesting to lift this restriction, but, to this end, one needs an intertwiner of several MacMahon representations, which have not yet been described. In [13], we found that the DIM R-matrix R λµ (x) for the Fock representations appears in the difference equation satisfied by the Nekrasov factor N λµ (x); It is an interesting problem to solve a generalization of (6.1) with R λµ (x) replaced by the MacMahon R-matrix R K 1 ,K 2 ΠΛ (x). The solution should be regarded as a generalized Nekrasov factor N K 1 ,K 2 ΠΛ (x). It is natural to expect that the factor is related to a norm of the triple Macdonald functions [25].
Another interesting aim might be the generalization of our story to the case of MacMahon modules with nontrivial asymptotics, i.e. to 3d Young diagrams with nontrivial "ends" along the coordinate axes [10,11]. We think one can use the same strategy by making a judicious choice of the spectral parameters of constituent Fock representations. Thus, the formulas should be quite similar to those presented above.
Finally, one can try to build the most general representation of the DIM algebra, in which the both central charges (not just one, as in the MacMahon case) are completely arbitrary. To our knowledge, such representations have never been studied, so it might lead to some unexpected surprises.

A Conventions and useful functions
The intriguing triality of the quantum toroidal algebra U q,d ( gl 1 ) becomes manifest, when we use the parameters (q 1 , q 2 , q 3 ) with q 1 q 2 q 3 = 1. These parameters are also natural from the point of view of plane partitions used in the MacMahon representation. In this paper, we use the following convention: 11 q 1 = dq −1 = q, q 2 = d −1 q −1 = t −1 , q 3 = q 2 = t/q. (A.1) The parameters (q, t) can be identified with those appear in the Macdonald functions. The formulas for the vertex operator representation can be expressed neatly by introducing the following factor: The function derived from the structure function g(z, w) = (z − q 1 w)(z − q 2 w)(z − q 3 w) of U q,d ( gl 1 ) appears in the formula of the MacMahon representation. Starting from the fundamental rational function we can reconstruct the function G(x) as follows: ψ(z) = ψ(z)ψ(q −1 2 z) −1 , G(z) =ψ(z)ψ(q −1 1 z) −1 . in the OPE relation of the vertex operators. 11 The parameters q2 and q3 are exchanged, compared with [26] and [11].

JHEP04(2019)097
The intertwiner Ψ(w) with the vertical Fock representation of central charges K 1 = 1, K 2 = t q and the spectral parameter w changes both the spectral parameter and the central charge of the horizontal representation. To account for this, we introduce the operators Q 1 and e 2 canonically conjugate to P 1 , and c 2 : The intertwiner is then given by where Φ λ (w) contains only nonzero bosonic modes. It is easy to derive the commutation relations of the intertwiners with the generators, as well as between the intertwiners themselves from the canonical commutation relations Q 1 P 1 = P 1 (Q 1 + 1), (C.9) c 2 e 2 = e 2 (c 2 + 1). (C.10) Notice that only part of the zero modes in eqs. (C.8) depends on the diagram living in the vertical Fock module: the essential part (conjugate to P 1 and c 2 ) is independent of the diagram.
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