The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies
- 10.2991/jnmp.k.210330.001How to use a DOI?
- Boson-Fermion correspondence; orthogonal and symplectic schur functions; integrable hierarchy; universal character
In this paper, we first construct an integrable system whose solutions include the orthogonal Schur functions and the symplectic Schur functions. We find that the orthogonal Schur functions and the symplectic Schur functions can be obtained by one kind of Boson-Fermion correspondence which is slightly different from the classical one. Then, we construct a universal character which satisfies the bilinear equation of a new infinite-dimensional integrable orthogonal UC hierarchy.
- © 2021 The Authors. Published by Atlantis Press B.V.
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Boson-Fermion correspondence is well-known in mathematical physics [1,11]. Young diagrams and symmetric functions are of interest to many researchers and have many applications in mathematics including combinatorics and representation theory [3,10]. There are many relations between Boson-Fermion correspondence and symmetric functions.
The KP hierarchy  is one of the most important integrable hierarchies and it arises in many different fields of mathematics and physics such as enumerative algebraic geometry, topological field and string theory. Schur functions have close relations with the τ -functions of KP hierarchy. Schur functions give the characters of finite-dimensional irreducible representations of the general linear groups, see [3,10]. Schur functions can be realized from vertex operators as in Equation (6) of this paper, and these vertex operators can be used to construct Fermions which act on Bosonic Fock space, see [7,11]. By replacing nxn by power sum, we find that the character of Young diagram in  is the same with the Schur function obtained from the Jacobi-Trudi formula, which tells us that the Schur functions are solutions of differential equations in the KP hierarchy, and the linear combinations of Schur functions with coefficients satisfying some relations (plücker relations) are also τ -functions of the KP hierarchy. In [12,13], the author generalized the KP hierarchy to the UC (universal character) hierarchy, whose τ -functions include universal characters .
The orthogonal and symplectic Schur functions are upgraded from Schur functions in the same setting . Symplectic Schur functions are equal to orthogonal Schur functions with the conjugate Young diagrams. Like Schur functions, the symplectic and orthogonal Schur functions can also be realized from vertex operators as in Equation (19), and these vertex operators can also be used to construct Fermions. Then there certainly exists an integrable system. In this paper, we will construct this integrable system, and show that the symplectic and orthogonal Schur functions are its solutions.
This paper is arranged as follows. In Section 2, we will recall the definition of Schur function, its vertex operator realization, and the relations between Schur functions and KP hierarchy. In Section 3, we will recall the definitions of orthogonal and symplectic Schur functions, their respective vertex operator realization, then we will define an integrable system whose τ -function can be obtained from orthogonal and symplectic Schur function. In Section 4, we will construct a method to calculate orthogonal and symplectic Schur functions from a different kind of Boson-Fermion correspondence. In Section 5, we will construct the modified type of the integrable system which is constructed in Section 3. In Section 6, we will consider the universal character and the corresponding UC hierarchy.
2. SCHUR FUNCTIONS, VERTEX OPERATOR AND THE KP HIERARCHY
Let x = (x1, x2, ⋯). The operators hn(x) are determined by the generating function:
For Young diagrams λ = (λ1, λ2, ⋯, λl), the Schur function Sλ = Sλ(x) is a polynomial in ℂ[x] defined by the Jacobi-Trudi formula :
Introduce the following vertex operators
Introduce Fermions and ψj for any as operators satisfying the relations
The Fock representation space of Fermions is the space of Maya diagrams. A Maya diagram is made up of black and white stones lined up along the real line with the convention that all the stones are black far away to the right, whereas all the stones are white far away to the left. For example, the following is a Maya diagram
By writing half integers u1, u2, ⋯ for the positions of the black stones, a Maya diagram is described as an increasing sequence of half integers
For example, the Maya diagram in (8) is denoted by
Define the charge p of a Maya diagram as the number of white stones on the right half line minus the number of black stones on the left half line. For example, the charge of Maya diagram in (8) is zero.
Let ℱ be the vector space based by the set of Maya diagrams, which is called Fermionic Fock space. The basis vector is written as |u〉. In particular,
The action of Fermions ψj and for any on Maya diagrams |u〉 is determined by the formulas
There are three vector spaces which are isomorphic to each other : the polynomial ring ℂ[x]= ℂ [x1, x2, ⋯] of infinitely many variables x = (x1, x2, ⋯) which is called the Bosonic Fock space, the charge zero part of the Fermionic Fock space ℱ, and the vector space Y spanned by Young diagrams. Therefore, the Maya diagram |u〉 can be written as
Let f(z, x) ∈ C[z, z−1, x1, x2, ⋯]. Define operators
Define the generating functions [5,11]
It can be checked that
that is, the operators determine a representation of the algebra spanned by Fermions, see equations in (7).
3. THE ORTHOGONAL SCHUR FUNCTION, THE SYMPLECTIC SCHUR FUNCTION, VERTEX OPERATORS AND AN INTEGRABLE HIERARCHY
For a Young diagram λ = (λ1, ⋯, λl), the orthogonal Schur function[6,9] is defined to be
Observe that this vertex operator VO(z) is the same as Vπ(z) for π = (2) in .
The operator is a raising operator of the orthogonal Schur function, i.e.,
Define the generating functions
It can be checked that
For an unknown function τ = τ (x), the bilinear equation
Equation (23) is equivalent to
It is obvious that equation (24) can be rewritten as
Let us replace (x′, x) with (x + u, x − u) and consider the Taylor series expansion at x′ = x, i.e., expand with respect to u = (u1, u2, ⋯). Hence, we obtain
By taking the coefficient of , we get many bilinear equations. Taking the coefficient of 1 = u0, we get
For a Young diagram λ = (λ1, ⋯, λl), the symplectic Schur function[6,9] is defined to be
The operator is a raising operator of the symplectic Schur function, i.e.,
For an unknown function τ = τ(x), the bilinear equation
4. ORTHOGONAL TYPE BOSON-FERMION CORRESPONDENCE
For Maya diagrams |u〉 and |v〉, the pairing 〈v|u〉 is defined by the formula
Define operators Hn by
From the actions of Fermions on Maya diagrams, we get the action of H1 on a Maya diagram is H1 sending a Maya diagram |u〉 to the sum over all Maya diagrams which can be obtained from |u〉 by moving a black stone to the right. We define Pn and Qn from equations
The action of Q(m) on Maya diagram is defined by Q(m) sending the Maya diagram |u〉 to the sum over all Maya diagrams which can be obtained from |u〉 by moving black stones m times to the right and no one black stone is moved twice. Then, Q(1m) sends Maya diagram |u〉 to the sum over all Maya diagrams which can be obtained from |u〉 by moving black stones m times to the right and no two adjacent black stones move at the same time.
The actions of , where , on Maya diagram can be obtained from the actions of ψj, and Q(m), Q(1m) on Maya diagram according to (34–35).
Let λ be a Young diagram, and λ′ be its conjugate. The Frobenius notation λ = (n1, ⋯, nl|m1, ⋯, ml) describes the Young diagram λ by ni = λi − i, , where l is the number of the boxes in the NW-SE diagonal line of λ.
Under the Boson-Fermion correspondence, the basis vector
For , the orthogonal Schur function is obtained from
Using the Fermions ψj and , we can also get the orthogonal Schur function by the following formula.
For , the orthogonal Schur function is obtained from
We can get the symplectic Schur function similarly.
For , the symplectic Schur function is obtained from
For example, we can calculate in the following two ways. The first way is
In the second way, we know that for the Maya diagram
Then, we obtain the orthogonal type Boson-Fermion correspondence.
The Fermions are realized in the Bosonic Fock space by , i.e., for any |u〉∈ ℱ, we have
5. THE MODIFIED ORTHOGONAL KP HIERARCHY
Now, we consider the functional relations for a sequence of τ -functions connected by successive application of vertex operators. Let τ0 ≔ τ(x) be a solution of the orthogonal KP hierarchy (23). Let τ1 ≔ VO(α)τ and with an arbitrary constant α ∈ ℂ×. Then τ1 and are also solutions of (23). Moreover, we can deduce the bilinear equation
Replace (x′, x) with (x + u, x − u) and consider the Taylor series expansion at x′ = x, we obtain
By taking the coefficient of for variety n, we will get many bilinear equations. Taking the coefficient of 1 = u0, we get
6. UNIVERSAL CHARACTER AND UC HIERARCHY
For a pair of Young diagrams λ = (λ1, ⋯, λl) and μ = (μ1, ⋯, μl′), we define the universal character as a polynomial in x = (x1, x2, ⋯) and y = (y1, y2, ⋯):
We can see that the orthogonal Schur function is a special case of the universal character: .
Let us introduce the vertex operators
It can be checked that the satisfy the formionic relations: and . The same relations hold also for . Moreover, and mutually commute.
The universal character can be obtained by means of these operators:
Proof. We will use the Vandermonde-like identity,
Taking the coefficient of , we will get (50).
We give a remark here to explain the difference between the universal characters here and that in our paper . The vertex operators which realize in this paper are more complex than that in , but in this paper, the universal characters can be described by the determinant, that in  can not described by determinant.
Now we can define a UC hierarchy where UC is the abbreviation of universal character.
For an unknown function τ = τ(x, y), the system of bilinear relations
If τ = τ(x, y) does not depend on y = (y1, y2, ⋯), the second equality of (51) trivially holds and the first one is reduced to the bilinear expression (23) of the OSKP hierarchy. From this aspect, we treat the orthogonal UC hierarchy as an extension of the OSKP hierarchy.
It is obvious that (51) can be rewritten into the form
Taking the coefficient of unvm leads to many differential equation with respect to x, y, these differential equations are all of infinite order. This reflects that the integrands above with (x = x′, y = y′) may be singular not only at z = 0, w = 0 but also at z = ∞, w = ∞.
In the follows, we give a class of polynomial solutions of the orthogonal UC hierarchy. From the relations between , we obtain
All the universal characters are solutions of the orthogonal UC hierarchy.
It is known that if τ = τ(x, y) is a solution of (51), so are X+(α)τ and Y+(β)τ for arbitrary constants α, β ∈ ℂ×. Then we will consider the bilinear relations among the solutions connected by the vertex operators. The modified orthogonal UC hierarchy is introduced as follows.
Suppose τm,n = τm,n(x, y) is a solution of the orthogonal UC hierarchy (51). Let
For τ - function τm,n = τm,n(x, y), the modified orthogonal UC hierarchy includes the following bilinear equations:
Here the first equation and the second equation can be deduced from (51) by applying 1 ⊗ X+(αm) and X+(αm) ⊗ Y+(βn), respectively.
From the definition of τm,n, for a solution τ0,0 of the orthogonal UC hierarchy, we have
For integers m, n ≥ 0, it holds that
The results above are obtained by applying , and to (51).
Let us look closely at (59), which corresponds to the orthogonal UC hierarchy (51) when m = n = 0. It can be equivalently rewritten into
Let I, J ⊂ ℤ be a disjoint pair of finite indexing sets. By specializing the parameters in (62) and (63) as
Let z = 1/w, we find that
Consequently, the integrands of (62) and (63) coincide up to constant functor if the condition |I|− |J|= m + n + 4 holds. Let
This means that the residue calculus at possible essential singularities z = 0, ∞ is avoided for the presence of two bilinear equations (62) and (63).
For a function f = f(x, y), we define a shift operator Ti by
The following equations hold:
If |I|− |J|= m + n + 4 and m, n ≥ 0, then
If |I|− |J|= n + 3 and n ≥ 0, then
If |I|− |J|= m + 3 and m ≥ 0, then
Let m = 1, n = 0, I = 1, 2, 3, 4, J = ∅, and let , the first equation in Proposition 6.6 reduces to
CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.
The authors gratefully acknowledge the support of Professors Ke Wu, Zi-Feng Yang and Shi-Kun Wang. This work is supported by the National Natural Science Foundation of China under Grant No. 11505046.
Cite this article
TY - JOUR AU - Linjie Shi AU - Na Wang AU - Minru Chen PY - 2021 DA - 2021/04/07 TI - The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies JO - Journal of Nonlinear Mathematical Physics SP - 292 EP - 302 VL - 28 IS - 3 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.k.210330.001 DO - 10.2991/jnmp.k.210330.001 ID - Shi2021 ER -