3D Bosons, 3-Jack polynomials and affine Yangian of ${\mathfrak{gl}}(1)$

3D (3 dimensional) Young diagrams are a generalization of 2D Young diagrams. In this paper, We consider 3D Bosons and 3-Jack polynomials. We associate three parameters $h_1,h_2,h_3$ to $y,x,z$-axis respectively. 3-Jack polynomials are polynomials of $P_{n,j}, n\geq j$ with coefficients in $\mathbb C(h_1,h_2,h_3)$, which are the generalization of Schur functions and Jack polynomials to 3D case. Similar to Schur functions, 3-Jack polynomials can also be determined by the vertex operators and the Pieri formulas.


Introduction
2D Young diagrams and symmetric functions are attractive research objects, which were used to determine irreducible characters of highest weight representations of the classical groups [1,2,3]. Recently they appear in mathematical physics, especially in integrable models. 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 β-deformed 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 3D symmetric functions.
The Schur functions S λ defined on 2D Young diagrams λ can be determined by the vertex operator and the Jacobi-Trudi formula. Let p = (p 1 , p 2 , · · · ). The operators S n (p) are determined by the vertex operator: e ξ(p,z) = ∞ n=0 S n (p)z n , with ξ(p, z) = ∞ n=1 p n n z n (1) and set S n (p) = 0 for n < 0. Note that S n (p) is the complete homogeneous symmetric function by the Miwa transform, i.e., replacing p i with the power sum k x i k . For 2D Young diagram λ = (λ 1 , λ 2 , · · · , λ l ), the Schur function S λ = S λ (p) is a polynomial of variables p in C[p] defined by the Jacobi-Trudi formula [1,2]: The Jacobi-Trudi formula can be replaced by the Pieri formula [1] S n S λ = µ C µ n,λ S µ .
The 3-Jack polynomials, which are symmetric functions defined on 3D Young diagrams, can also be determined this way. We associate three parameters h 1 , h 2 , h 3 to y, x, z-axis respectively, where h 1 , h 2 , h 3 are the parameters appeared in affine Yangian of gl(1). 3-Jack polynomials are symmetric about three coordinate axes, which means they are symmetric about h 1 , h 2 , h 3 . Let (n) be the 2D Young diagrams of n box along y-axis, which is treated as 3D Young diagram which have one layer in z-axis direction, then 3-Jack polynomialsJ n can be determined by the vertex operator: e ξy(P,z) = n≥0 1 J n ,J n h n 1J n (P )z n , with ξ y (P, z) = ∞ n,j=1 where d n,j = 1 if n = j, j if n > j.
When P n,1 = p 1 , P n,j≥1 = 0, the vertex operator (4) becomes (1). The 3-Jack polynomials of n boxes along x-axis or z-axis can also expressed this way from the symmetry. In fact, for 2D Young diagrams λ, which are treated as the 3D Young diagrams who have one layer, 3-Jack polynomialsJ λ can also be determined by the vertex operators. 3-Jack polynomials of 3D Young diagrams who have more than one layer can be determined by the "Pieri formula"J λJπ with π being 3D Young diagrams. This Pieri formula can be determined by the representation of affine Yangian of gl(1) on 3D Young diagrams. 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 consider 3D Bosons and the algebra W . In section 4, we discuss the realizations of 3D Bosons and the operators in the algebra W by using affine Yangian of gl(1) and its representation on 3D Young diagrams. In section 5, we show the expressions of 3-Jack polynomials by the vertex operators and the Pieri formulas.
2 Affine Yangian of gl (1) In this section, we recall the definition of the affine Yangian of gl(1) as in papers [11,12,13,14] first. Then we calculate some properties of affine Yangian which have relations with 3D Bosons. 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 [13,14] [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. In this paper, we set ψ 0 = 1 with no loss of generality. The notations σ 2 , σ 3 in the definition of affine Yangian are functions of three complex numbers h 1 , h 2 and h 3 : This affine Yangian has the representation on 3D Young diagrams or Plane partitions. A plane partition π is a 2D Young diagram in the first quadrant of plane xOy filled with non-negative integers that form nonincreasing rows and columns [15,9]. The number in the position (i, j) is denoted by π i,j   π 1,1 π 1,2 · · · π 2,1 π 2,2 · · · · · · · · · · · ·   .
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 [16], we use the following notations. For a 3D Young diagram π, the notation ✷ ∈ 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 ✷ ∈ 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 ✷, we let where (x ✷ , y ✷ , z ✷ ) is the coordinate of box ✷ in coordinate system O − xyz. Here we use the order y ✷ , x ✷ , z ✷ to match that in paper [13]. Following [13,14], 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 [13] 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 (23) and (24) mean generators e j , f j acting on the 3D Young diagram π by The triangular decomposition of affine Yangian Y is where Y + is the subalgebra generated by generators e j with relations (8) and (15), B is the commutative subalgebra with generators ψ j , Y − is the subalgebra generated by generators f j with relations (9) and (16). Define the anti-automorphismã bỹ The quadratic form on Y + |0 is defined bỹ where x, y ∈ Y + . Note that the quadratic form here is different from the Shapovalov form in [13]. For 3D Young diagrams π, π ′ and let π = x|0 , π ′ = y|0 for x, y ∈ Y + , define the orthogonality π ′ , π = 0|ã(y)x|0 .
In the following, we calculate some properties of affine Yangian which have close relation with 3D Bosons. We want to have the fact that 3D Bosons will become 2D Bosons when we require a n,k = 0, k > 1 and a n,1 = a n . As in paper [13], when n > 0, a n,1 and a −n,1 are defined to be a n, a −n,1 := 1 (n − 1)! ad n−1 e 1 e 0 .

3D Bosons
Introduce the space of all polynomials Every polynomial is a function of infinitely many variables but each polynomial itself is a finite sum of monomials, so involves only finitely many of the variables. Define the weight of p n,k to be n, and the weight of monomial p l 1 n 1 ,k 1 · · · p ls ns,ks to be l 1 n 1 + · · · + l s n s , then the 3D Bosonic Fock space is written into 3 , we can write the basis of every subspace C[p] n . By the results in [17], we know that the elements in the basis of C[p] n is in one to one correspondence with 3D Young diagrams of total box number n. Then we have where the notation dim(C[p] n ) means the dimension of the vector space C[p] n . Define the form ·, · on C[p] by p n,j , p m,i = 0 unless n = m, i = j. When i = j = 1, p n,1 , p m,1 equal p n , p m in the 2D Bosons. Define and Consider operators b n,k , where n ∈ Z, n = 0, and k ∈ Z, 0 < k ≤ |n|, with the commutation relations The operators b n,k with the relations above are called 3D Boson, and the algebra generated by b n,k with relations (38) is called 3D Heisenberg algebra, which is denoted by B. Using the commutation relations (38), we can see that any element in 3D Heisenberg algebra can be expressed in a unique way as a linear combination of the following elements: where the notation (m, l) < (n, k) means m < n or m = n, l < k.
Define a linear map ρ : for n > 0, which gives a representation of the Heisenberg algebra B on polynomial space C[p]. The representation space C[p] is called the 3D Bosonic Fock space. We call the operators b n,k annihilation operators and b −n,k creation operators for n > 0. From the commutation relations (38), it is easy to find that all the creation operators commute among themselves, so do all the annihilation operators. The element 1 ∈ C[p] is called the vacuum state. Every annihilation operators kill the vacuum state, that is, n ∂ ∂p n,k 1 = 0. The 3D Bosonic Fock space is generated by the vacuum state: We give a remark to explain why we use "3D" in the 3D Boson and 3D Heisenberg algebra. 3D is used to distinguish 3D Boson and ordinary Boson, 3D Heisenberg algebra and ordinary Heisenberg algebra. Similar to that the ordinary Bosonic Fock space is isomorphic to the space of 2D Young diagrams, the 3D Bosonic Fock space is isomorphic to the space of 3D Young diagrams, this is the reason we use the notation 3D.
In the following, we consider the algebra W with the generators a n,k , (n ∈ Z, n = 0, and k ∈ Z, 0 < k ≤ |n|). We let a n,1 = b n,1 and a −2, [a n,1 , a m,j ] = 0, when j > 1, and when j > 1, k > 1, [a m,j , a n,k ] = 0≤l≤j+k−2 j+k−leven where the coefficients N l jk (m, n) are and the structure constants C l jk are with (a) n = a(a + 1) · · · (a + n − 1), [a] n = a(a − 1) · · · (a − n + 1), We give a remark here to explain the central charges. The central charge of a n,1 is ψ 0 , we always take ψ 0 = 1 without loss the symmetry of affine Yangian of gl(1) about three coordinate axes. The central charge c j of a n,j is dependent on j. The first few of them are . Define which match P 2,2 , P 2,2 and P 3,3 , P 3,3 in [11] respectively, and which matches P 4,4 , P 4,4 in [18]. Here we choose P 4,4 = E 13 |0 in [18]. For n > 0, define P n,j be the representation of a −n,j on 3D Young diagrams, then the 3-Jack polynomials as vectors in the representation space are functions of P n,j . Note that the set {P n,j } are related with {p n,j }. For example, P n,1 = p n,1 and P 2,2 = p 2,2 .
We denote the first term in a n,2 by 2L n , and the second term by 2L n . Then we have that L n andL n separately satisfy the virasoro relations, that is, By calculation, we can prove the following relations hold From (13) and (14), we have [ψ 2 , a n,1 ] = −2na n,1 for any n. Then we have Proof. When n < 0, it holds clearly. When n > 0, it can be proved by induction.
Proposition 4.6. For n ≥ 1, Proof. Since the relation (57) hold, we only need to prove where (L −n ) + include all terms which have annihilation operators inL −n , that is, Then which means the result holds.
Then P n 2,2 , P n 2,2 = n! n j=1 For n ≥ j, we also have P n,j , P n,j = n + j − 1 n − j P j,j , P j,j . (72)

3-Jack polynomials
In this section, we want to obtain the expression of 3-Jack polynomialsJ π for any 3D Young diagram π. It is known that Schur functions S λ can be determined by for λ = (λ 1 , λ 2 , · · · , λ 2 ). The formula (74) is equivalent to the Pieri formula S n S λ = µ C µ n,λ S µ , where the Pieri formula can be found in [2]. For example, S n,1 = det S n S n+1 1 S 1 from (74), which is the same with the Pieri formula S 1 S n = S n+1 + S n,1 .
We require the 3-Jack polynomials become the symmetric functions Y λ under these conditions.

Concluding remarks
In this paper, all results are obtained by requiring ψ 0 = 1. If interested, one can calculate the results for general ψ 0 , which should be similar to that in this paper. For example, in this paper should be for general ψ 0 .
This holds since there is the scaling symmetries in the affine Yangian of gl(1) [13]. The scaling symmetries say that changing the value of ψ 0 is equivalent to rescaling parameters h j , j = 1, 2, 3. Next, we will consider the slice of 3-Jack polynomials similar to that the slices of 3D Young diagrams are 2D Young diagrams.

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.