UC and BUC plane partitions
This paper is concerned with the investigation of UC and BUC plane partitions based upon the fermion calculus approach. We construct generalized the vertex operators in terms of free charged fermions and neutral fermions and present the interlacing (strict) 2-partitions. Furthermore, it is showed that the generating functions of UC and BUC plane partitions can be written as product forms.
Keywords: 17B80; 35Q55; 37K10
Introduction
Free charged fermions and neutral fermions proposed by Kyoto school [[1]–[5]] play a crucial role in construction of -functions of integrable systems such as KP and BKP hierarchies. Tsuda [[6]] introduced the universal character (UC) hierarchy which is the generalization of KP hierarchy. Then Ogawa [[7]] constructed UC hierarchy of B-type (BUC hierarchy) which can be regarded as the extension of BKP hierarchy. The algebraic structures of UC and BUC hierarchies have been well discussed based upon the free fermions and neutral fermions [[6]–[8]]. By means of fermion calculus [[5]], the relations between vertex operators and KP plane partitions have been developed [[9]]. Fermionic approach is a extremely useful tool in exploring the structure and properties of integrable systems. Ünal [[10]] presented the -functions of the KP and BKP hierarchies as determinants and Pfaffians with charged free fermions and neutral free fermions.
Plane partitions are generated in crystal melting model [[12]] which have widely applications in various fields of mathematics and physics, such as statistical models, number theory and representation theory. The generating function of plane partitions describes the characteristics of plane partitions which has widely application in combinatorics [[14]], statistical mechanics [[16]] and integrable systems [[18]]. Okounkov et al. [[19]] analyzed generating function for plane partitions in terms of vertex operators expressed as exponentials of bilinear in fermions. Then the partition functions of the topological string theory have been developed by the fermion calculus approach [[20]]. Recently, Wang et al. [[21]] investigated 3-dimensional (3D) Boson representation of algebra and studied Littlewood-Richardson rule for 3-Jack polynomials by acting 3D Bosons on 3D Young diagrams (plane partitions). By using the fermion calculus approach, Foda et al. [[9], [23]] established the product forms for the generating function of KP and BKP plane partitions based on the KP free charge fermions and BKP neutral fermions, respectively. It is also proved the generating function is a special -function of the 2D Toda lattice. The aim of this paper is to investigate the generating function of plane partitions for UC and BUC hierarchies.
The paper is organized as follows. Section 2 provides a review of the fundamental facts of free fermions, plane partitions and generating functions. Section 3 is devoted to investigation of the UC plane partitions by fermion calculus approach. We introduce the interlacing partitions are presented with half-integers and construct interlacing 2-partitions, from which a product form of the generating function for UC plane partitions are derived. In Sect. 4, By introducing generating interlacing strict 2-partitions, we study the generating function for BUC plane partitions. The last section is conclusions and discussions.
Preliminaries
In this section, we mainly retrospect basic facts about free fermions, plane partitions and generating functions [[5], [23]–[26]].
Charged fermions and UC hierarchy
and are charged fermions, the charge of the fermions is given by
Fermion |
|
|
|
|
---|
Charge | (1, 0) |
| (0, 1) |
|
Algebra over is generated by the commutation relations
2.1
Graph
and
A Maya diagram is made up of black and white stones lined up along the real line, indexed by half-integers. It is required that far away to the right (when ) all the stones are black, whereas far away to the left (when they are all white. By writing for the position of the black stone, we can describe a Maya diagram as an increasing sequence of half-integers
2.2
Graph
that satisfies the following conditions
2.3
Graph
The right state corresponding to the Maya diagram is defined as
2.4
Graph
A left action of the fermions is given by the following rules
2.5
Graph
2.6
Graph
In particular,
2.7
Graph
2.8
Graph
Similarly, a Maya diagram can also be represented as
2.9
Graph
where denotes the position of the white stone and for all sufficiently large j. The left state corresponding to the Maya diagram is denoted as
2.10
Graph
A right action of the fermions is given by
2.11
Graph
2.12
Graph
while and have respectively the same action as and except replacing with Particularly, the vacuum state and the dual vacuum state are defined as
2.13
Graph
which satisfy
2.14
Graph
The charged fermionic Fock space and the dual Fock space are generated by
2.15
Graph
where
2.16
Graph
The vector subspace of with charge is written as Consider a pairing denoted by
2.17
Graph
where is called the vacuum expectation value. The following properties hold
2.18
Graph
The UC hierarchy is a system satisfying the following bilinear identity
2.19
Graph
where has charge (0, 0).
Define the colon operator : : as
2.20
Graph
Consider the operators and
2.21
Graph
Then the following properties hold
2.22
Graph
Noting
2.23
Graph
The operators called Hamiltonian are defined as
2.24
Graph
along with the generating functions of charged fermions
2.25
Graph
For convenience, is represented as
Proposition 2.1
The commutative relations between the Hamiltonian and generating functions of charged fermions are as follows
2.26
Graph
where
2.27
Graph
Proof
By means of Eqs. (2.22) and (2.24), we obtain
2.28
Graph
The other formulas can be proved in the same way.
Lemma 2.2
The following equations hold
2.29
Graph
Proof
From the Eq. (2.26) , it follows that
2.30
Graph
Using the similar procedure, we can prove other equations.
Remark 2.3
Under the reduction Eq. (2.19) leads to bilinear identity of KP hierarchy. The Eqs. (2.1)–(2.29) leads to definitions and properties in KP hierarchy.
Neutral fermions and BUC hierarchy
In this section, we introduce neutral fermions and which are generators of the algebra over and satisfy
2.31
Graph
The neutral fermionic Fock space and the dual Fock space can be defined as
2.32
Graph
where the vacuum state and the dual vacuum state are denoted by
2.33
Graph
Introduce the operators and
2.34
Graph
Note that
2.35
Graph
In particular,
2.36
Graph
The Hamiltonian is written as
2.37
Graph
It should be noted that BUC hierarchy satisfies the bilinear identity, which is given in [[7]].
Lemma 2.4
For the generating functions of neutral fermions,
2.38
Graph
we have
2.39
Graph
where
2.40
Graph
Proof
From Eqs. (2.35) and (2.37), we obtain
2.41
Graph
Therefore, we have
2.42
Graph
The proof of the other formula is quite similar, so is omitted.
Consider the neutral fermion vertex operators
2.43
Graph
where
2.44
Graph
Plane partitions
A partition (strict partition) is a non-increasing (strictly decreasing) sequence consisting of non-negative integers, denoted as with weights Define a partition which is obtained by taking the transpose of Suppose that there are r nodes on the main diagonal of partitions and set for we have The partition can be also expressed as
2.45
Graph
A hook refers to the set of boxes
2.46
Graph
The partition can be denoted by the hook as
2.47
Graph
where and
Example 2.5
The partition in Fig. 1 can be constructed by a set of hooks, where and
Graph: Fig. 1 Partition α=(4,2,0|3,1,0)
For the partitions and we say that interlaces and write which is defined by the following relation
2.48
Graph
where and Consider the set
2.49
Graph
where and All partitions that intersect and are contained in
Let a strict partitions the right state and left state corresponding to can be written as
2.50
Graph
where
2.51
Graph
Lemma 2.6
[[9]] Setting from Eqs. (2.43) and (2.50), the following relations hold
2.52
Graph
2.53
Graph
Graph: Fig. 2 A 3-dimensional view of a plane partition π. The value of πij denotes the number of boxes stacked at the location
Graph: Fig. 3 A 2-dimensional view of the plane partition in Fig. 2. The sequence of values covered in the slice is the corresponding partition. In particular, π0=(4,4,1) and |π|=31
Graph: Fig. 4 A 2-dimensional view of a strict plane partition π~. The sequence of values covered in the slice are strictly decreasing. The difference between Fig. 4 and Fig. 3 is the main diagonal
A plane partition is a set of non-negative integers which satisfies
2.54
Graph
Each plane partition can be represented as a composition of specific partitions, denoted as Indicate as
2.55
Graph
then the plane partition satisfies
2.56
Graph
for sufficiently large and the weight
A strict plane partition satisfies
2.57
Graph
for all integers
For a strict plane partition we refer to the set of all connected boxes as paths, which are connected horizontal plateaux in the 3-dimensional view. denotes the number of paths possessed by For strict plane partitions and represents the number of nonzero elements in and represents the number of non-zero elements in but not in
The generating function for plane partitions is given by
2.58
Graph
The generating function for strict plane partitions can be expressed as
2.59
Graph
UC plane partitions
In this section, we construct generalized charged fermion vertex operators and investigate interlacing 2-partitions. By means of fermion calculus approach, the generating function for UC plane partitions has been developed.
Generalized charged fermion vertex operators
Introduce
3.1
Graph
where z and v are indeterminate. The Hamiltonian can be rewritten as
3.2
Graph
Let us define the generalized charged fermion vertex operators
3.3
Graph
It is easy to derive
3.4
Graph
Taking and we have
3.5
Graph
Proposition 3.1
The vertex operators and satisfy the following relations
3.6
Graph
Proof
We only prove the first formula of Eq. (3.6), other formulas can be proved similarly. By means of Eqs. (2.26), (2.29) and (3.5), we get
3.7
Graph
It follows from Eq. (2.25) that
3.8
Graph
Comparing the orders of k on both sides yields
3.9
Graph
The operators and satisfy
Graph
Then
3.10
Graph
Remark 3.2
The vertex operators and are reduced to the charged fermion vertex operators and by deleting the variables and v, respectively. Then Eqs. (3.4)–(3.10) lead to the properties for KP hierarchy.
Generating interlacing 2-partitions
If the Maya diagram has charge 0, there is a one-to-one correspondence between the Maya diagram and the partition. The right state corresponding to partitions and in space can be represented as
3.11
Graph
where and The left state has a similar representation in the charge (0, 0) sector of the dual Fock space
3.12
Graph
where and
Define 2-partition and write which represents a pair of partitions and Then we have and the weight Let 2-partitions and we say that interlaces and write
3.13
Graph
In particular, if 2-partition is reduced to the partition Equations (3.11) and (3.12) lead to
3.14
Graph
Definition 3.3
An 'UC plane partition' is defined as which denotes a pair of plane partitions and satisfies
3.15
Graph
where the weight of the UC plane partition is the sum of the weights of these 2-partitions.
Example 3.4
The UC plane partition in Fig. 5 represents a pair of plane partitions and where and the weight is
Graph: Fig. 5 The UC plane partition (χ-3,χ-2,χ-1,χ0,χ1,χ2,χ3,χ4)
Lemma 3.5
Let and be states corresponding to the partition in the Fock space and the dual Fock space which are described in Eq. (3.14). Then we have
3.16
Graph
3.17
Graph
Proof
Set From Eqs. (3.6) and (3.14), one obtains
3.18
Graph
where
3.19
Graph
The following equation holds
3.20
Graph
Using the commutation relations (2.1), we have
3.21
Graph
From Eqs. (2.14), (3.20) and (3.21), we obtain
3.22
Graph
Therefore
3.23
Graph
Set
3.24
Graph
It can be clearly found that the terms of the expansion of the Eq. (3.23) contain all of the partitions in accompanied by the weighting factor z. Each weighted partition can be expressed as
3.25
Graph
The powers of z can be written as
3.26
Graph
where
3.27
Graph
From Eqs. (3.24)–(3.27), the Eq. (3.23) can be rewritten as
3.28
Graph
A similar proof for the left state yields
3.29
Graph
For let
3.30
Graph
Then one obtains
3.31
Graph
Lemma 3.6
Let the states corresponding to the 2-partition be and The following relations hold
3.32
Graph
3.33
Graph
Proof
By means of the Eq. (3.11)
3.34
Graph
where
3.35
Graph
By using the Eq. (3.16), we have
3.36
Graph
Setting is rewritten as
3.37
Graph
According to the commutation relations (2.1) one obtains
3.38
Graph
where
3.39
Graph
Taking combining the Eq. (3.11) gets
3.40
Graph
Using the similar approach yields
3.41
Graph
Setting Eqs. (3.40) and (3.41) are reduced to
3.42
Graph
The case of is similar to the above.
Generating function for UC plane partitions
Consider the correlation function
3.43
Graph
where p and q are indeterminate. Set 2-partition and insert in the middle of a pair of multiplicative vertex operators. It follows that
3.44
Graph
By means of Eqs. (3.40)–(3.42), the generated weights are of the form
3.45
Graph
Set and then we have
3.46
Graph
where and Note that the plane partition consists of and the plane partition consists of Hence interlacing relation above indicates
3.47
Graph
The Eq. (3.45) can be rewritten as
3.48
Graph
Then we derive the generating function for UC plane partitions
3.49
Graph
On the other hand, by using the Eqs. (3.4) and (3.10), we can express the generating function for UC plane partitions as the product of the generalized MacMahon's formula
3.50
Graph
BUC plane partitions
In this section, the BUC plane partitions will be developed. By using the fermion calculus method, we construct generalized neutral fermion vertex operators. Based upon interlacing strict 2-partitions derived by the vertex operator, we investigate the properties of the generating function for BUC plane partitions.
Generalized neutral fermion vertex operators
Set
4.1
Graph
where z and v are indeterminate. Replacing the variables above, we obtain
4.2
Graph
Meanwhile the generalized neutral fermion vertex operators and are defined as
4.3
Graph
In particular,
4.4
Graph
Taking the transformation of and we have
4.5
Graph
Proposition 4.1
The following equations hold
4.6
Graph
Proof
From Eqs. (2.39), (2.41) and (4.5), it is clear that
4.7
Graph
Substituting Eq. (2.38) into the above equation and comparing the orders of k, one obtains
4.8
Graph
Other equations can be proved with the same method.
By means of Eqs. (2.35) and (4.2), we have
4.9
Graph
It follows that
4.10
Graph
Generating interlacing strict 2-partitions
Let strict partitions and In the Fock space and the dual Fock space the states corresponding to and can be described as
4.11
Graph
where and
4.12
Graph
Denote strict 2-partition as which possesses the same properties as 2-partition. Note that if strict 2-partition is equivalent to the strict partition Eq. (4.11) leads to
4.13
Graph
Definition 4.2
Define the 'BUC plane partition' as which represents a pair of BKP plane partitions and where and
Lemma 4.3
Let the states corresponding to the strict 2-partition be and Then
4.14
Graph
4.15
Graph
where and
4.16
Graph
Proof
By means of Eqs. (4.6) and (4.11), one obtains
4.17
Graph
where
4.18
Graph
Substituting Eq. (2.52) into Eq. (4.18), we have
4.19
Graph
Since the assumed state is not involved in the subsequent calculations, we let
4.20
Graph
From the commutation relations (2.31), the Eq. (4.17) can be rewritten as
4.21
Graph
It follows from Eq. (2.52) that
4.22
Graph
Setting
4.23
Graph
Applying the above results to Eq. (4.21) yields
4.24
Graph
Similarly, it is show that
4.25
Graph
In particular, if Eqs. (4.24) and (4.25) are respectively transformed into
4.26
Graph
A similar conclusion can be obtained for
Generating function for BUC plane partitions
Define correlation function as
4.27
Graph
which provides a generating function for BUC plane partitions, where t and q are indeterminate.
Graph: Fig. 6 All the paths in Fig. 4
Proposition 4.4
For a strict plane partition we have
4.28
Graph
Proof
Let us use the example of strict plane partition in Fig. 4 to explain this formula. From Fig. 4, it is clear that
4.29
Graph
and
4.30
Graph
From the above relations, it is showed that diagonal slices not being intersected receive a factor of 2, otherwise zero. Multiplying as the power of 2 and in Fig. 6, we have
4.31
Graph
It follows that this method extends to arbitrary strict plane partitions.
Setting a strict 2-partition and inserting to the above equation yields
4.32
Graph
Equations (4.24)–(4.26) show that the generated weights are given by
4.33
Graph
Let and Note that the plane partition is made up of and the plane partition is made up of Combining Proposition 4.4, the Eq. (4.33) can be represented as
4.34
Graph
It shows that all strict 2-partitions satisfy
4.35
Graph
which is equivalent to
4.36
Graph
Then the Eq. (4.34) can be rewritten as
4.37
Graph
It follows that
4.38
Graph
In addition, by means of the Eq. (4.10), the generating function for BUC plane partitions can be represented as
4.39
Graph
Equation (4.39) can be regarded as the extension of the shifted MacMahon's formula [[26]].
Conclusions and discussions
In this paper, by means of constructing the generalized fermion vertex operators and interlacing (strict) 2-partitions, we have discussed generating functions for UC and BUC plane partitions which can be written as product forms. It should be pointed out that the fermion calculus approach play a vital role in establishing generating functions of plane partitions. How to use this method to look for the structure and properties of plane partitions in other integrable systems, such as symplectic universal character (SUC) hierarchy and the orthogonal universal character (OUC) should be an interesting question, which will be studied in the near future.
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11965014 and 12061051), the Natural Science Foundation of Inner Mongolia Autonomous Region (Grant No. 2023MS01003) and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT23096). The authors gratefully acknowledge the support of Professor Ke Wu and Professor Weizhong Zhao at Capital Normal University, China.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: This paper is devoted to connecting "Universal character" with plane partition. The study focus on the mathematical construction and computation. That is why there is no data.]
Code availability
My manuscript has no associated code/software. [Author's comment: Code/Software sharing not applicable to this article as no code/software was generated or analysed during the current study.]
[
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]
By Shengyu Zhang and Zhaowen Yan
Reported by Author; Author