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Volume 28, Issue 3, September 2021, Pages 292 - 302

The Orthogonal and Symplectic Schur Functions, Vertex Operators and Integrable Hierarchies

Authors
Linjie Shi, Na Wang*, Minru Chen
School of Mathematics and Statistics, Henan University, Kaifeng, 475001, China
*Corresponding author: Email: wangna@henu.edu.cn
Corresponding Author
Na Wang
Received 18 March 2018, Accepted 10 March 2021, Available Online 7 April 2021.
DOI
10.2991/jnmp.k.210330.001How to use a DOI?
Keywords
Boson-Fermion correspondence; orthogonal and symplectic schur functions; integrable hierarchy; universal character
Abstract

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.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

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 [1] 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 [11] 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 [8].

The orthogonal and symplectic Schur functions are upgraded from Schur functions in the same setting [2]. 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:

eξ(x,k)≔∑n=0∞hn(x)kn,  where ξ(x,k)=∑n=1∞xnkn (1)
and set hn(x) = 0 for n < 0. Note that if we replace ixi with the power sum pi=∑nxni, hn (x) is the complete homogeneous symmetric function [10]
∑i1≤i2≤⋯≤inxi1xi2⋯xin. (2)

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 [8]:

Sλ(x)=det (hλi-i+j(x))1≤i,j≤l. (3)

Introduce the following vertex operators

V+(k)=∑n∈𝕑Vn+kn=eξ(x,k)e-ξ(∂˜x,z-1), (4)
V-(k)=∑n∈𝕑Vn-kn=e-ξ(x,k)eξ(∂˜x,k-1). (5)
where ∂˜x=(∂x1,12∂x2,⋯,1n∂xn,⋯). The operators Vi+ are raising operators for the Schur functions
Sλ(x)≔Vλ1+⋯Vλl+⋅1 (6)
where λ is a Young diagram (λ1, λ2, ⋯, λl), and we denote Sλ(x) by Sλ for short.

Introduce Fermions ψj* and ψj for any j∈𝕑+12 as operators satisfying the relations

{ψj,ψk}=0,{ψj*,ψk*}=0,{ψj*,ψk}=δj+k,0 (7)
where {A, B} = AB + BA. The generating functions of Fermions are
ψ(k)≔∑j∈𝕑+1/2ψjk-j-1/2,ψ*(k)≔∑j∈𝕑+1/2ψj*k-j-1/2.

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

⋯○○●●○●○●●⋯-72-52-32-121232527292 (8)

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

u={un}n≥1  with  u1<u2<u3<⋯.

For example, the Maya diagram in (8) is denoted by

-32,-12,32,72,92,⋯

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,

|0〉=|12,32,52,⋯〉.

The action of Fermions ψj and ψj* for any j∈12+𝕑 on Maya diagrams |u〉 is determined by the formulas

ψj|u〉={(-1)i-1|⋯,ui-1,ui+1,⋯〉if ui=-j for some i,0otherwise, (9)
ψj*|u〉={(-1)i|⋯,ui,j,ui+1,⋯〉if ui<j<ui+1for some i,0otherwise. (10)

There are three vector spaces which are isomorphic to each other [11]: 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

|u〉=|λ,n〉=|Sλ,n〉,
where n is the charge of |u〉. In the special case of n = 0, we also write the Maya diagram |u〉 as |λ〉.

Let f(z, x) ∈ C[z, z−1, x1, x2, ⋯]. Define operators

eKf(z,x)≔zf(z,x),kH0f(z,x)≔f(kz,x). (11)

Define the generating functions [5,11]

V˜(k)≔∑j∈𝕑+12V˜jk-j-12=V+(k)eKkH0, (12)
V˜*(k)≔∑j∈𝕑+12V˜j*k-j-12=V-(k)eKkH0. (13)

It can be checked that

{V˜i,V˜j}=0,{V˜i*,V˜j*}=0,{V˜i,V˜j*}=δi+j,0, (14)

that is, the operators V˜i,V˜j* determine a representation of the algebra spanned by Fermions, see equations in (7).

Definition 2.1.

For an unknown function τ = τ (x), the bilinear equation

∑j∈𝕑+12V˜j*τ⊗V˜-jτ=0 (15)

is called the KP hierarchy, see [5,11].

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

SλO≔det(hλi-i+j-hλi-i-j)1≤i,j≤l, (16)
where hn is the nth complete symmetric function of the form in equation (2). Define vertex operators
VO(z)≔(1-z2)eξ(x,z)e-ξ(∂˜x,z-1)e-ξ(∂˜x,z) (17)
VO*(z)≔e-ξ(x,z)eξ(∂˜x,z-1)eξ(∂˜x,z) (18)
and let
VO(z)=∑n∈𝕑VnOzn,  VO*(z)=∑n∈𝕑VnO*zn.

Observe that this vertex operator VO(z) is the same as Vπ(z) for π = (2) in [2].

The operator VnO is a raising operator of the orthogonal Schur function, i.e.,

SλO(x)=Vλ1OVλ2O⋯VλlO⋅1 (19)
for a partition λ = (λ1, λ2, ⋯, λl).

Define the generating functions

XO(k)=∑j∈𝕑+12XjOk-j-12=VO(k)eKkH0, (20)
XO*(k)=∑j∈𝕑+12XjO*k-j-12=VO*(k)eKkH0. (21)

It can be checked that

{XiO,XjO}=0,{XiO*,XjO*}=0,{XiO,XjO*}=δi+j,0. (22)

Definition 3.1.

For an unknown function τ = τ (x), the bilinear equation

∑j∈𝕑+12XjO*τ⊗X-jOτ=0 (23)
is called the orthogonal/symmplectic KP hierarchy, and denoted by OSKP hierarchy for short.

Equation (23) is equivalent to

∑n+m=-1VnO*τ⊗VmOτ=0. (24)

It is obvious that equation (24) can be rewritten as

12πi∮(1-z2)eξ(x-x′,z)dzτ(x′+[z-1]+[z])τ(x-[z-1]-[z])=0 (25)
with x = (x1, x2, ⋯) and x′=(x1′,x2′,⋯) being arbitrary parameters. Here the symbol [z] denotes (z,z22,z33,⋯) and the integration means taking the coefficient of 1z of the integrand in the formal Laurent series expansion in z. Then the equation (25) is equivalent to
Res(1-z2)eξ(x-x′,z)τ(x′+[z-1]+[z])τ(x-[z-1]-[z])=0. (26)

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

∑i-j+k=-1Pi(-2u)Pj(∂˜u)Pk(∂˜u)τ(x+u)τ(x-u)-∑i-j+k=-3Pi(-2u)Pj(∂˜u)Pk(∂˜u)τ(x+u)τ(x-u)=0. (27)

By taking the coefficient of un=u1n1u2n2⋯, we get many bilinear equations. Taking the coefficient of 1 = u0, we get

∑k=0∞Pk+1(Dx)Pk(Dx)τ(x)⋅τ(x)-∑k=0∞Pk+3(Dx)Pk(Dx)τ(x)⋅τ(x)=0, (28)
where Dx=(Dx1,12Dx2,13Dx3,⋯). We see that every differential equation with respect to x contained in the orthogonal KP hierarchy is of infinite order. This reflects the fact that the integrand of (25) with x′ = x may be singular not only at z = 0, but also at z = ∞.

For a Young diagram λ = (λ1, ⋯, λl), the symplectic Schur function[6,9] is defined to be

SλSp=12det (hλi-i+j+hλi-i-j+2)1≤i,j≤l,
where hn is the nth complete symmetric function. The Symplectic symmetric function can be obtained by vertex operators as follows. Define the vertex operators
VSp(z)=eξ(x,z)e-ξ(∂x,z-1)e-ξ(∂x,z) (29)
VSp*(z)=(1-z2)e-ξ(x,z)eξ(∂x,z-1)eξ(∂x,z) (30)
and let
VSp(z)=∑n∈𝕑VnSpzn,  VSp*(z)=∑n∈𝕑VnSp*zn,
here the vertex operator VSp(z) is the same as Vπ(z) for π = (12) in [2].

The operator VnSp is a raising operator of the symplectic Schur function, i.e.,

SλSp(x)=SλSp(x)=Vλ1SpVλ2Sp⋯VλlSp⋅1 (31)
for a partition λ = (λ1, λ2, ⋯, λl).

For an unknown function τ = τ(x), the bilinear equation

∑n+m=-1VnSp*τ⊗VmSpτ=0 (32)
gives the same integrable system as the bilinear equation (23), that is why we call this integrable system OSKP hierarchy.

4. ORTHOGONAL TYPE BOSON-FERMION CORRESPONDENCE

For Maya diagrams |u〉 and |v〉, the pairing 〈v|u〉 is defined by the formula

〈v|u〉=δv1+u1,0δv2+u2,0⋯.

Define operators Hn by

Hn=∑j∈𝕑+1/2:ψ-jψj+n*:
and H(x)=∑n=1∞xnHn.

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

exp(∑m≥1Hmmkm)=∑n≥0Q(n)kn,  exp(∑m≥1H-mmkm)=∑n≥0P(n)kn (33)

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.

Define

ψjO=∑n=1∞(-1)nψn+jQ1n-∑n=1∞(-1)nψn+j+2Q1n, (34)
ψjO*=∑n=1∞ψn+jQn. (35)

The actions of ψjO,ψjO*, where j∈12+𝕑, on Maya diagram can be obtained from the actions of ψj, 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, mi=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

ψn1⋯ψnlψm1*⋯ψml*|vac〉 for n1<⋯<nl<0 and m1<⋯<ml<0
of Fermionic Fock space of charge zero goes over into the Schur function Sλ multiplied by aλ=(-1)∑i=1l(mi+12)+l(l-1)2, where λ=(-n1-12,⋯,-nl-12|-m1-12,⋯,-ml-12), i.e.,
Sλ=aλ〈vac|eH(x)ψn1⋯ψnlψm1*⋯ψml*|vac〉, (36)
then we have

Proposition 4.1.

For λ=(-n1-12,⋯,-nl-12|-m1-12,⋯,-ml-12), the orthogonal Schur function SλO is obtained from

SλO=(-1)∑i=1l(mi+12)+l(l-1)2〈vac|eH(x)ψn1O⋯ψnlOψm1O*⋯ψmlO*|vac〉. (37)

Using the Fermions ψj and ψj*, we can also get the orthogonal Schur function by the following formula.

Proposition 4.2.

For λ=(-n1-12,⋯,-nl-12|-m1-12,⋯,-ml-12), the orthogonal Schur function SλO is obtained from

SλO=(-1)∑i=1l(mi+12)+l(l-1)2〈vac|eH(x)e-∑n=1∞12n(Hn2+H2n)ψn1⋯ψnlψm1*⋯ψml*|vac〉. (38)

We can get the symplectic Schur function similarly.

Proposition 4.3.

For λ=(-n1-12,⋯,-nl-12|-m1-12,⋯,-ml-12), the symplectic Schur function SλSp is obtained from

SλSp=aλ〈vac|eH(x)ψn1Sp⋯ψnlSpψm1Sp*⋯ψmlSp*|vac〉 (39)
=aλ〈vac|eH(x)e-∑n=1∞12n(Hn2-H2n)ψn1⋯ψnlψm1*⋯ψml*|vac〉, (40)
where aλ=(-1)∑i=1l(mi+12)+l(l-1)2.

For example, we can calculate S◫O in the following two ways. The first way is

ψ-32O=ψ-32-ψ12-ψ-12Q+ψ32Q+⋯,ψ-12O*=ψ-12*+ψ12*Q+ψ32*Q2+⋯,

Then,

S◫O=〈vac|eH(x)ψ-32Oψ-12O*|vac〉=〈vac|eH(x)(ψ-32-ψ12)ψ-12*|vac〉=S◫-S0.

In the second way, we know that for the Maya diagram

|γ〉=⋯○○○●●○●●●⋯-72-52-32-121232527292
we have Hmγ = 0 when m > 2. Then
S◫O=〈vac|eH(x)e-∑n=1∞12n(Hn2+H2n)ψ-32ψ-12|vac〉=〈vac|eH(x)(1-12(H12+H2))|γ〉=〈vac|eH(x)(1-Q2)|γ〉=S◫-S0.

Then, we obtain the orthogonal type Boson-Fermion correspondence.

Proposition 4.4.

The Fermions ψjO,ψjO* are realized in the Bosonic Fock space by XjO,XjO*, i.e., for any |u〉∈ ℱ, we have

〈l|eH(x)ψjO|u〉=XjO〈l|eH(x)|u〉, 〈l|eH(x)ψjO*|u〉=XjO*〈l|eH(x)|u〉, (41)
where 〈l|=〈⋯,l-52,l-32,l-12|.

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 τ1′=VSp(α)τ with an arbitrary constant α ∈ ℂ×. Then τ1 and τ1′ are also solutions of (23). Moreover, we can deduce the bilinear equation

∑n+m=-2VnO*τn⊗VnOτn+1=0
from (23) multiplied by1 ⊗ VO(α) or
∑n+m=-2VnSp*τn⊗VnSpτn+1=0
from (32) multiplied by1 ⊗ VSp(α). The two equations above can be equivalently rewritten into the equation
12πi∮z(1-z2)eξ(x-x′,z)dzτn(x′+[z-1]+[z])τn+1(x-[z-1]-[z])=0. (42)

Replace (x′, x) with (x + u, x − u) and consider the Taylor series expansion at x′ = x, we obtain

∑i-j+k=-2Pi(-2u)Pj(∂˜u)Pk(∂˜u)τn(x+u)τn+1(x-u)+∑i-j+k=-4Pi(-2u)Pj(∂˜u)Pk(∂˜u)τn(x+u)τn+1(x-u)=0. (43)

By taking the coefficient of un=u1n1u2n2⋯ for variety n, we will get many bilinear equations. Taking the coefficient of 1 = u0, we get

∑k=0∞Pk+2(Dx)Pk(Dx)τn(x)⋅τn+1(x)-∑k=0∞Pk+3(Dx)Pk(Dx)τn(x)⋅τn+1(x)=0. (44)

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, ⋯):

S[λ,μ]O(x,y)=(-1)ll′+l′(l′+1)2×det(hμl′-i+1-l-l′+i-j-1(y)-hμl′-i+1-l-l′+i+j-1(y),1≤i≤l′hλi-l′+l′-i+j(x)-hλi-l′+l′-i-j(x),l′+1≤i≤l+l′). (45)

We can see that the orthogonal Schur function SλO(x) is a special case of the universal character: SλO(x)=S[λ,∅]O(x,y).

Let us introduce the vertex operators

X+(k)=(1-k2)eξ(x,k)e-ξ(∂˜y,k-1)e-ξ(∂˜y,k)e-ξ(∂˜x,k-1)e-ξ(∂˜x,k), (46)
X-(k)=e-ξ(x,k)eξ(∂˜y,k-1)eξ(∂˜y,k)eξ(∂˜x,k-1)eξ(∂˜x,k), (47)
Y+(k)=(1-k2)eξ(y,k)e-ξ(∂˜x,k-1)e-ξ(∂˜x,k)e-ξ(∂˜y,k)e-ξ(∂˜y,k-1), (48)
Y-(k)=e-ξ(y,k-1)eξ(∂˜x,k-1)eξ(∂˜x,k)eξ(∂˜y,k)eξ(∂˜y,k-1), (49)
and let X±(k)=∑n∈𝕑Xn±kn, Y±(k)=∑n∈𝕑Yn±kn.

It can be checked that the Xn± satisfy the formionic relations: Xn±Xm+1±+Xn+1±Xn±=0 and X n+1+X m-+X m+1-X n+=δn+m+1,0. The same relations hold also for Yn±. Moreover, Xn± and Yn± mutually commute.

Proposition 6.1.

The universal character S[λ,μ]O(x,y) can be obtained by means of these operators:

S[λ,μ]O(x,y)=Xλ1+⋯Xλl+Yμ1+⋯Yμl+⋅1. (50)

Proof. We will use the Vandermonde-like identity,

det(kil-j-kil+j)=∏1≤i<j≤l(ki-kj)(1-kikj).

Then,

X+(k1)⋯X+(kl)Y+(w1-1)⋯Y+(wl′-1)⋅1  =∏i=1l(1-ki2)∏j=1l′(1-1wi2)∏i<j(1-kikj)(1-kjki)∏i,k(1-1kiwj)(1-kiwj)  ×∏a<b(1-wbwa)(1-1wawb)eξ(x,k1)⋯eξ(x,kl)eξ(y,w1-1)⋯eξ(y,wl′-1)  =(-1)ll′+l′(l′+1)2∏ikl-i-(l+2l′-i)∏jwl′-i+1-(2l+2l′-i+1)det(wl′-i+1l+l′-j-wl′-i+1l+l′+j,1≤i≤l′ki-l′l+l′-j-ki-l′l+l′+j,l′+1≤i≤l+l′)  ×eξ(x,k1)⋯eξ(x,kl)eξ(y,w1-1)⋯eξ(y,wl′-1)  =(-1)ll′+l′(l′+1)2det(wl′-i+1-l-l′+i-j-1-wl′-i+1-l-l′+i+j-1,1≤i≤l′ki-l′-l′+i-j-ki-l′-l′+i+j,l′+1≤i≤l+l′)  ×eξ(x,k1)⋯eξ(x,kl)eξ(y,w1-1)⋯eξ(y,wl′-1).

Taking the coefficient of k1λ1⋯klλlw1-μ1⋯wl′-μl, we will get (50).

We give a remark here to explain the difference between the universal characters S[λ,μ]O(x,y) here and that in our paper [4]. The vertex operators which realize S[λ,μ]O(x,y) in this paper are more complex than that in [4], but in this paper, the universal characters S[λ,μ]O(x,y) can be described by the determinant, that in [4] can not described by determinant.

Now we can define a UC hierarchy where UC is the abbreviation of universal character.

Definition 6.2.

For an unknown function τ = τ(x, y), the system of bilinear relations

∑n+m=-1Xn-τ⊗Xm+τ=∑n+m=-1Yn-τ⊗Ym+τ=0 (51)
is called the orthogonal UC hierarchy.

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

12πi∮(1-z2)eξ(x′-x″,z)dz  τ(x″+[z-1]+[z],y″+[z])⋅τ(x′-[z-1]-[z],y′-[z])=0, (52)
12πi∮(1-w2)eξ(y′-y″,w)dw  τ(x″+[w],y″+[w-1]+[w])⋅τ(x′-[w],y′-[w-1]-[w])=0 (53)
for arbitrary x, x′, y and y′. Consider their Taylor expansions at (x = x′, y = y′), that is, replacing (x, x′, y, y′) with (x − u, x + u, y − v, y + v) and expand with respect to (u, v) = (u1, u2, ⋯, v1, v2, ⋯), then we get
∑i-j+k-m+n=-1Pi(-2u)Pj(∂˜u)Pk(∂˜u)Pm(∂˜u)Pn(∂˜u)τ(x+u,y+v)τ(x-u,y-v)-∑i-j+k-m+n=-3Pi(-2u)Pj(∂˜u)Pk(∂˜u)Pm(∂˜u)Pn(∂˜u)τ(x+u,y+v)τ(x-u,y-v)=0.
and
∑i-j+k-m+n=-1Pi(-2v)Pj(∂˜u)Pk(∂˜u)Pm(∂˜u)Pn(∂˜u)τ(x+u,y+v)τ(x-u,y-v)-∑i-j+k-m+n=-3Pi(-2v)Pj(∂˜u)Pk(∂˜u)Pm(∂˜u)Pn(∂˜u)τ(x+u,y+v)τ(x-u,y-v)=0.

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 Xn±,Yn±, we obtain

(∑n+m=-1Xn-⊗Xm+)(Xt+⊗Xt+)=(Xt+1+⊗Xt-1+)(∑n+m=-1Xn-⊗Xm+) (54)
(∑n+m=-1Yn-⊗Ym+)(Xt+⊗Xt+)=(Xt+⊗Xt+)(∑n+m=-1Yn-⊗Ym+) (55)
that is, if τ = τ (x, y) is a solution of (51), so is Xt+τ, we can verify in the same way that Yt+τ is also a solution of (51). By equation (50), we obtain

Proposition 6.3.

All the universal characters S[λ,μ]O(x,y) 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.

Definition 6.4.

Suppose τm,n = τm,n(x, y) is a solution of the orthogonal UC hierarchy (51). Let

τm+1,n=X+(αm)τm,n,τm,n+1=Y+(βn)τm,n,τm+1,n+1=X+(αm)Y+(βn)τm,n=Y+(βn)X+(αm)τm,n
for arbitrary constants αm, βn ∈ ℂ×. From equation (51), we can get the equations satisfied by τm,n’s, which are called the modified orthogonal UC hierarchy.

For τ - function τm,n = τm,n(x, y), the modified orthogonal UC hierarchy includes the following bilinear equations:

∑i+j=-2Xi-τm,n⊗Xj+τm+1,n=∑i+j=-1Yi-τm,n⊗Yj+τm+1,n=0, (56)
τm,n⊗τm+1,n+1-∑i+j=0Xi-τm+1,n⊗Xj+τm,n+1=∑i+j=-2Yi-τm+1,n⊗Yj+τm,n+1=0. (57)

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

τm,n=∏i=0m-1X+(αi)∏j=0n-1Y+(βj)τ0,0, (58)
where
∏i=0m-1X+(αi)=X+(αm-1)⋯X+(α1)X+(α0),
then we have the following bilinear equations.

Proposition 6.5.

For integers m, n ≥ 0, it holds that

∑i+j=-m-1Xi-τ0,0⊗Xj+τm,n=∑i+j=-n-1Yi-τ0,0⊗Yj+τm,n=0, (59)
τ0,0⊗τ1,n-∑i+j=0Xi-τ1,0⊗Xj+τ0,n=∑i+j=-n-1Yi-τ1,0⊗Yj+τ0,n=0, (60)
∑i+j=-m-1Xi-τ0,1⊗Xj+τm,0=τ0,0⊗τm,1-∑i+j=0Yi-τ0,1⊗Yj+τm,0=0. (61)

The results above are obtained by applying 1⊗∏i=0m-1X+(αi)∏j=0n-1Y+(βj), X+(α0)⊗∏j=0n-1Y+(βj) and Y+(β0)⊗∏i=0m-1X+(αi) 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

12πi∮zm(1-z2)eξ(x-x′,z)dz  τ0,0(x′+[z-1]+[z],y′+[z-1]+[z])⋅τm,n(x-[z-1]-[z],y-[z-1]-[z])=0, (62)
12πi∮wn(1-w2)eξ(y-y′,w)dw  τ0,0(x′+[w-1]+[w],y′+[w-1]+[w])⋅τm,n(x-[w-1]-[w],y-[w-1]-[w])=0. (63)

Let I, J ⊂ ℤ be a disjoint pair of finite indexing sets. By specializing the parameters in (62) and (63) as

x′=x-∑j∈I[ti]+∑j∈J[tj],y′=y-∑j∈I[ti-1]+∑j∈J[tj-1],
we get
Ω1≔zm(1-z2)eξ(x-x′,z)dz=zm(1-z2)∏j∈J(1-tjz)∏j∈I(1-tjz)dz,Ω2≔wn(1-w2)eξ(x-x′,w)dw=wn(1-w2)∏j∈J(1-w/tj)∏j∈I(1-w/tj)dw.

Let z = 1/w, we find that

Ω2=z|I|-|J|-m-n-4∏j∈I(-tj)∏j∈J(-tj)Ω1

Consequently, the integrands of (62) and (63) coincide up to constant functor if the condition |I|− |J|= m + n + 4 holds. Let

F(z)=zm(1-z2)∏j∈J(1-tjz)∏j∈I(1-tjz)⋅τ(x′+[z-1]+[z],y′+[z-1]+[z])⋅τ(x-[z-1]-[z],y-[z-1]-[z])
in the integrand of (62), hence, we get
∫C1F(z)dz=∫C2F(z)dz=0,
where C1 and C2 are a positively oriented small circle around z = 0 and z = ∞ respectively such that all the other singularities are out of it. Then, we obtain
∑i∈IResz=1/tiF(z)dz=0. (64)

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

Ti(f)=f(x-[ti],y-[ti-1])
and T{i1,⋯,ir} ≔ Ti1 ⋯ Tir(f) for short. Then (64) gives
∑i∈Itin(1-1ti2)∏j∈J(ti-tj)∏j∈I/{i}(ti-tj)TI/{i}τ0,0(x+[ti-1],y+[ti])TJ∪{i}τm,n(x-[ti-1],y-[ti])=0,
which can be regarded as a difference equation with each ti being the difference interval. Then, we have

Proposition 6.6.

The following equations hold:

  1. 1.

    If |I|− |J|= m + n + 4 and m, n ≥ 0, then

    ∑i∈Itin(ti2-1)∏j∈J(ti-tj)∏j∈I/{i}(ti-tj)TI/{i}τ0,0(x+[ti-1],y+[ti])TJ∪{i}τm,n(x-[ti-1],y-[ti])=0.

  2. 2.

    If |I|− |J|= n + 3 and n ≥ 0, then

    TI(τ0,0)TJ(τ1,n)=∑i∈I(1-1ti2)∏j∈J(1-tj/ti)∏j∈I/{i}(1-tj/ti)TI/{i}τ1,0(x+[ti-1],y+[ti])⋅TJ∪{i}τ0,n(x-[ti-1],y-[ti])=0.

  3. 3.

    If |I|− |J|= m + 3 and m ≥ 0, then

    TI(τ0,0)TJ(τm,1)=∑i∈I(1-1ti2)∏j∈J(1-ti/tj)∏j∈I/{i}(1-ti/tj)TI/{i}τ1,0(x+[ti-1],y+[ti])⋅TJ∪{i}τ0,n(x-[ti-1],y-[ti])=0.

Let m = 1, n = 0, I = 1, 2, 3, 4, J = ∅, and let t3=t1-1,t4=t2-1, the first equation in Proposition 6.6 reduces to

(1-t1t2)(t2-t1)T˜12(τ0,0)τ1,1=t2(t12+1)T˜2(τ1,0)T˜1(τ0,1)-t1(t22+1)T˜1(τ1,0)T˜2(τ0,1), (65)
where the notation T˜i is a shift operator defined by
T˜if(x,y)=f(x-[ti]-[ti-1],y-[ti]-[ti-1]).

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENTS

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.

REFERENCES

[1]E Date, M Kashiwara, M Jimbo, and T Miwa, Transformation groups for soliton equations, M Jimbo and T Miwa (editors), Nonlinear integrable systems- classical theory and quantum theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, pp. 39-119.
[3]W Fulton and J Harris, Representation theory, A first course, Springer-Verlag, New York, 1991.
[10]IG Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 1979.
[11]T Miwa, M Jinbo, M Jimbo, and E Date, Solitons: differential equations, symmetries and infinite dimensional algebras, Cambridge University Press, Cambridge, 2000.
[12]T Tsuda, From KP/UC hierarchies to Painlevé equations. arXiv:1004.1347
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Journal of Nonlinear Mathematical Physics
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Publication Date
2021/04/07
ISSN (Online)
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risenwbib
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  -