3D Bosons and $W_{1+\infty}$ algebra

In this paper, we consider 3D Young diagrams with at most $N$ layers in $z$-axis direction, which can be constructed by $N$ 2D Young diagrams on slice $z=j$, $j=1,2,\cdots, N$ from the Yang-Baxter equation. Use 2D Bosons $\{a_{j,m},\ m\in\Z\}$ associated to 2D Young diagrams on the slice $z=j$, we constructed 3D Bosons. Then we show the 3D Boson representation of $W_{1+\infty}$ algebra, and the Littlewood-Richardson rule for 3-Jack polynomials from the actions of 3D Bosons on 3D Young diagrams.


Introduction
The Schur functions defined on 2D Young diagrams are an attractive research object, which were used to determine irreducible characters of highest weight representations of the classical groups, and the Littlewood-Richarson rule for Schur functions show the relations between the representation spaces [1,2,3]. There are many structures, such as 2D Bosons and Boson-Fermion correspondence, defined on Schur functions or 2D Young diagrams. These structures have many applications in mathematical physics. In [4], the group in the Kyoto school uses Schur functions in a remarkable way to understand the KP and KdV hierarchies. In [5,6], Tsilevich and Su lkowski, respectively, give the realization of the phase model in the algebra of Schur functions and build the relations between the q-boson model and Hall-Littlewood functions. In [7], we build the relations between the statistical models, such as phase model, and KP hierarchy by using 2D Young diagrams and Schur functions. In [8], the authors show that the states in the βdeformed Hurwitz-Kontsevich matrix model can be represented as the Jack polynomials.
3D Young diagrams (plane partitions) are a generalization of 2D Young diagrams, which arose naturally in crystal melting model [9,10]. 3D Young diagrams also have many applications in many fields of mathematics and physics, such as statistical models, number theory, representations of some algebras (Ding-Iohara-Miki algebras, affine Yangian, etc). In this paper, we consider 3D Bosons and the Littlewood-Richardson rule for 3-Jack polynomials on 3D Young diagrams which parallel to 2D Bosons and the Littlewood-Richardson rule for Schur functions or Jack polynomials on 2D Young diagrams.
Let a j,n be the 2D Bosons associated to 2D Young diagrams which are on the slice z = j of 3D Young diagrams with the relation [a j,n , a i,m ] = − 1 h 1 h 2 δ i,j nδ n+m,0 , where h 1 , h 2 are the parameters in the affine Yangian of gl (1). 3D Bosons b n,j can be represented by these 2D Bosons. We treat 3D Young diagrams which have one layer in z-axis direction as 2D Young diagrams. Since we require 3D Bosons become 2D Bosons when N = 1, which means 3D Bosons b n,1 become 2D Bosons and b n,j≥2 become zero, we know that 0|b n,j≥2 b −n,j≥2 |0 must contain the factor 1 + h 1 h 2 ψ 0 , where ψ 0 = − N h 1 h 2 is the generator in affine Yangian of gl (1). Since all the results we constructed on 3D Young diagrams are symmetric about three coordinate axes, which means they are symmetric about the parameters h 1 , h 2 , h 3 , then 0|b n,j≥2 b −n,j≥2 |0 must contain the factor The Littlewood-Richardson rule for Schur functions are well known, for example, S (2) S (1,1) = S (3,1) + S (2,1,1) .
They show the relations between the irreducible representation spaces of the general linear groups or permutation groups. In this paper, we calculate the Littlewood-Richardson rule for 3-Jack polynomials, we will find that it is more complicated than that for Schur functions, but it will become that for Schur functions in the special case h 1 = 1, h 2 = −1, N = 1. We believe that the Littlewood-Richardson rule should have applications in representation theory which we will consider later.
The paper is organized as follows. In section 2, we recall the definition of affine Yangian of gl(1) and its representation on 3D Young diagrams. In section 3, we recall the definition of the W 1+∞ algebra, then we construct the fields in W 1+∞ algebra from the Miura transformation. The Virasoro field V 2 (z) become that in [11] when h 1 = h, h 2 = −h −1 . The spin 3 field V 3 (z) is given new. In section 4, we construct 3D Boson fields and give the 3D Boson representation of W 1+∞ algebra. In section 5, we give the Littlewood-Richardson rule for 3-Jack polynomials. In section 6, we show the actions of 3D Bosons on 3D Young diagrams and the relations between 3D Bosons and the generators of affine Yangian of gl(1).

Affine Yangian of gl(1)
In this section, we recall the definition of affine Yangian of gl(1) and its representation on 3D Young diagrams. The affine Yangian Y of gl(1) is an associative algebra with generators e j , f j and ψ j , j = 0, 1, · · · and the following relations [12,13] [ψ j , ψ k ] = 0, [e j , f k ] = ψ j+k , together with boundary conditions and a generalization of Serre relations where Sym is the complete symmetrization over all indicated indices which include 6 terms.
The integers π i,j satisfy for all integers i, j ≥ 0. Piling π i,j cubes over position (i, j) gives a 3D Young diagram. 3D Young diagrams arose naturally in the melting crystal model [9,10]. We always identify 3D Young diagrams with plane partitions as explained above. For example, the 3D Young diagram can also be denoted by the plane partition (1, 1). As in our paper [15], we use the following notations. For a 3D Young diagram π, the notation 2 ∈ π + means that this box is not in π and can be added to π. Here "can be added" means that when this box is added, it is still a 3D Young diagram. The notation 2 ∈ π − means that this box is in π and can be removed from π. Here "can be removed" means that when this box is removed, it is still a 3D Young diagram. For a box 2, we let where (x 2 , y 2 , z 2 ) is the coordinate of box 2 in coordinate system O − xyz. Here we use the order y 2 , x 2 , z 2 to match that in paper [12]. Following [12,13], we introduce the generating functions: where u is a parameter. Introduce and For a 3D Young diagram π, define ψ π (u) by In the following, we recall the representation of the affine Yangian on 3D Young diagrams as in paper [12] by making a slight change. The representation of affine Yangian on 3D Young diagrams is given by where |π means the state characterized by the 3D Young diagram π and the coefficients Equations (20) and (21) mean generators e j , f j acting on the 3D Young diagram π by In the following of this paper, we consider 3D Young diagrams which have at most N layers in the z-axis direction, and slice the 3D Young diagrams into a series of 2D Young diagrams by the plane z = n for n = 1, 2, · · · , N . Then the symmetry of the affine Yangian of gl(1) about the coordinate axes are broken. For example, ψ 0 = − N h 1 h 2 .

W 1+∞ algebra
We begin this section by the definition of W 1+∞ algebra. The W 1+∞ algebra contains the Heisenberg algebra, the Virasoro algebra as subalgebras [12]. The generators are V j,m for j ∈ Z + , m ∈ Z. The relations are [V 2,m , V 1,n ] = −nV 1,m+n , and generally where the coefficients N l jk (m, n) are and the structure constants C l jk are Note that here we allow that the central charges can be different. Define then the bosons a n satisfy In the following two subsections, we consider the Boson a j,n representation of W 1+∞ algebra.

Miura transformation and the
and define the operator U k (z) as in [11] by The fields U k (z) generate an algebra, which is W 1+N . The fields V n (z) can be realized by U k (z). We list the concrete expressions of the first few U k (z) as in [16] U 0 = 1, (39) Note that the expressions of U k (z) is the same with that in [16], but the commutation relation of Boson fields J j (z) are different from that in [16]. Clearly, the Boson field is the same with that in [16] with a slight different commutation relation (or OPE) which means the central charge of Boson field To get the Virasora field, we need the following OPE where we use AB(z) to denote the normal order : A(z)B(z) :. Note that these relations become that in [16] when which satisfy We also have which equals the special case h = h 3 in [18], and the following OPE with As in paper [11], let the central charge c 2 equals Note that the central charge c 2 here is the same with that in [16,11], but the expression of the stress-energy field V 2 (z) is different from T 1+∞ (z) in [16,11]. To obtain V 3 (z), we calculate which equals we have Note that here if we replace V 3 (w) by −V 3 (w), the minus before V 2 (w) in the OPE above will disappear. We use the expression of V 3 (z) in (51) since we want the coefficient of U 3 (z) in V 3 (z) to be −1, which matches the calculation in [16]. The OPEs and The central charge c 3 From (35) and (36), we know that DefineV The centerc 1 ofV 1 (z) is 1.
The OPEs related toV 4 (z) arē Note that when N = 1,

3D Bosons
In this section, we show the 3D Boson fields and 3D Boson representation of W 1+∞ algebra.

3D Boson fields
We denote 3D Boson fields by B j (z). Clearly, B 1 (z) = J(z). We found that B 2 (z) can not be V 2 (z) defined in (45) since the central charge of V 2 (z) is ψ 0 σ 2 + ψ 3 0 σ 2 3 which does not become zero when h 1 = h, h 2 = −1/h, ψ 0 = 1. We define which equals and satisfy with the central charge c B 2 = −2(1 + ψ 0 σ 2 + ψ 3 0 σ 2 3 ), which equals P 2,2 , P 2,2 in [17]. We notice that with Note that when N = 1, that is, 3D Young diagrams become 2D Young diagrams, the 3D Boson B 1 (z) becomes 2D Boson field J 1 (z), and the 3D Boson field B 2 (z) become zero. We define which satisfies From the calculations above, we see that the 3D Boson fields are Boson field U 1 (z) and the algebra W ∞ which is given in [16]. We found that when N = 1, the 3D Boson fields become 2D Boson field U 1 (z) = J 1 (z), which corresponds to that 3D Young diagrams become 2D Young diagrams, concretely, when N = 1, the 3D Boson field B 1 (z) becomes 2D Boson field U 1 (z) = J 1 (z), and the 3D Boson fields B j≥2 (z) become zero. These results match that when N = 1, the 3D variables P n,j≥2 become zero and 3-Jack polynomials become 2D symmetric functions Y λ (2D Jack polynomials when

3D Boson representation of the W 1+∞ algebra
In this subsection, we use 3D Boson fields to represent the fields V n (z) in W 1+∞ algebra. We see that V 1 (z) = B 1 (z). For n = 2, which can also be written as When n = 3, We also have

The Littlewood-Richardson rule for 3-Jack polynomials
In this section, we show the Littlewood-Richardson rule for 3-Jack polynomials. Define For j = 1, 2, · · · , N , the Boson algebra a j,n has an representation on 2D Young diagrams λ or the symmetric functions Y λ , where Y λ are defined in [18] and are functions of power sums p n , n = 1, 2, · · · . The symmetric functions Y λ become 2D Jack polynomials when h 1 = h, h 2 = −h −1 . We denote the power sums related to a j,n by p j,n . Denote the vacuum state by |0 j , the actions of a j,n on |0 j are a j,−n |0 j = p j,n |0 j , a j,n |0 j = − 1 h 1 h 2 ∂ ∂p j,n |0 j , n > 0.
3D Bosons are the operators acting on |0 . For n > 0, define We calculate P n,k in order to obtain its property. When k = 1, (a j,−l−1 a k,l + a k,−l−1 a j,l ),(82) then which equals P 2,2 calculated in [19]. Generally, When k = 3, and generally where j = 1, 2, · · · , N and m = 1, 2, · · · in p j,m . We can see that P n,j = 0 when n < j, which means that 3-Jack polynomials are functions of variables P n,j with n ≥ j. Define the degree of P n,j equal n, then the graded vector space of P n,j is isomorphic to that of 3D Young diagrams which is graded by the box numbers of 3D Young diagrams.
For 3D Young diagram , 3-Jack polynomialJ equals For 3D Young diagrams of two boxes, we havẽ where the first equation is equal to that in [17] when ψ 0 = 1, and the second equation is equal to that in [19]. Here what we want to do is constructing the Littlewood-Richardson rule for 3-Jack polynomials. We can see that the expressions of P n,j are much easier than that of b −n,j . From [19], we know the expressions of P n,j , from them we can not obtain the expressions of b −n,j . The 3D Bosons b −n,j are operators acting on 3D Young diagrams or 3-Jack polynomials, acting on the vacuum state |0 , b −n,j |0 = P n,j · 1, whereJ 0 = 1, but for any 3D Young diagram π, b −n,j |π = P n,j ·J π . This is the difference between 3D Bosons and 2D Bosons. For 2D Bosons b −n,1 and any 2D Young diagram λ, we have b −n,1 |λ = P n,1 ·J λ .

The representation of 3D Bosons on 3D Young diagrams
In this section, we show the representation of 3D Bosons on 3D Young diagrams or 3-Jack polynomials. Since 3-Jack polynomials are functions of variables p j,n with j = 1, 2, · · · , N and n = 1, 2, · · · , the actions of 3D Bosons b n,j on 3-Jack polynomialsJ π can be determined by a j,−n ·J π ({p j,n }) = p j,nJπ ({p j,n }), for n > 0.
Since we have known the representation of affine Yangian of gl(1) on 3D Young diagrams, in the following, we use the generators of affine Yangian of gl(1) to represent 3D Bosons. We know that [18] and Compare with the expression of b n,1 , we obtain that b −n,1 = 1 (n − 1)! ad n−1 e 1 e 0 , The expressions in (98) and (99) equals that in [18]. Let ψ 0 = 1 without loss the generality. For Boson field B 2 (z), we have which also explains b −1,2 |0 = 0 since e 1 |0 = 0. For n ≥ 1, which equal that in [20].
As operators, we have   We can see that the Littlewood-Richardson rule for 3-Jack polynomials are complicated. When h 1 = 1, h 2 = −1, the Littlewood-Richardson rule for 3-Jack polynomials becomes that for Schur functions: when h 1 = 1, h 2 = −1, 3-Jack polynomials of 3D Young diagrams which have more than one layer in z-axis direction become zero, and 3-Jack polynomials of 3D Young diagrams which have one layer in z-axis direction become the Schur functions of the corresponding 2D Young diagrams, for example, 3-Jack polynomi-alsJ We also can check that when h 1 = h, h 2 = −h −1 , the Littlewood-Richardson rule for J ({P n,j }) ×J ({P n,j }) becomes that for 2D Jack polynomialsJ (2)J(1,1) .

Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of interest statement
The authors declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.