# Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

Authors: Kohei Motegi
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We apply the Izergin-Korepin analysis to the study of the projected wavefunctions of the generalized free-fermion model. We introduce a generalization of theL-operator of the six-vertex model by Bump-Brubaker-Friedberg and Bump-McNamara-Nakasuji. We make the Izergin-Korepin analysis to characterize the projected wavefunctions and show that they can be expressed as a product of factors and certain symmetric functions which generalizes the factorial Schur functions. This result can be seen as a generalization of the Tokuyama formula for the factorial Schur functions.


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## Body

## 1. Introduction

Integrable lattice models [1–4] are special classes of models in statistical physics in which many exact calculations are believed to be able to be done. The most local object in integrable models is called the R-matrix, and its mathematical structure was revealed in the mid-1980s [5, 6]. The underlying mathematical structure was named as the quantum groups, and the investigation of the quantum groups naturally leads to immediate constructions of various R-matrices.From the point of view of statistical physics,R-matrices are the most local objects, and the study on the R-matrices is a starting point. The most important objects in statistical physics are partition functions. For the case of integrable models, partition functions are objects constructed from multiple R-matrices and are determined by boundary conditions. One of the most famous partition functions in integrable lattice models are the domain wall boundary partition functions which was first introduced and analyzed in [7, 8]. In recent years, a more general class of partition functions which we shall call as the projected wavefunctions are attracting attention in its relation with algebraic combinatorics. The projected wavefunctions are the projection of the off-shell Bethe vector of integrable models into a class of some simple states labelled by the sequences of the particles or down spins. For the case of the free-fermion model in an external field, it was first shown by Bump-Brubaker-Friedberg [9] that the projected wavefunctions give a natural realization of the Tokuyama combinatorial formula for the Schur functions [10], which is a one-parameter deformation of the Weyl character formula (note that there are pioneering works using the free-fermion model implicitly in [11–13], and the Izergin-Korepin analysis and observation of the factorization phenomena on the domain wall boundary partition functions of the related models are called the Perk-Schultz (supersymmetric vertex) model [14] and the Felderhof free-fermion model [15] in [16, 17]. There is also an application to the correlation functions in [18]). This observation triggered studies on finding various generalizations and variations of the Tokuyama-type formula for symmetric functions [19–27] such as the factorial Schur functions and symplectic Schur functions, and an interesting notion was introduced furthermore which the number theorists call it the metaplectic ice.In this paper, we analyze the free-fermion model using the method initiated by Izergin-Korepin [7, 8]. The method was developed by them in order to find the explicit expression of polynomials representing the domain wall boundary partition functions of the Uq(sl2) six-vertex model, from which the famous Izergin-Korepin determinant formula was found. The Izergin-Korepin analysis is the important method to study variants of the domain wall boundary partition functions. For example, it was applied to the domain wall boundary partition functions of the Uq(sl2) six-vertex model with reflecting end by Tsuchiya [28] to find its determinant formula. Extending the Izergin-Korepin analysis to more general class of partition functions is also important. Wheeler [29] invented a method to extend the Izergin-Korepin analysis on a class of partition functions called the scalar products. And in our very recent work [30], we extended the Izergin-Korepin analysis to study the projected wavefunctions of the Uq(sl2) six-vertex model. The resulting symmetric polynomials representing the projected wavefunctions contains the Grothendieck polynomials as a special case when the six-vertex model reduces to the five-vertex model [31–33]. We apply this technique to study the free-fermion model in an external field. To this end, we first introduce an ultimate generalization of the L-operator by introducing the inhomogeneous parameters and factorial parameters. We use an inhomogeneous version of the generalized L-operator in our forthcoming paper [34] having two types of factorial parameters, which generalizes the factorial L-operator by Bump-McNamara-Nakasuji [22]. We next view the projected wavefunctions as a function of the inhomogeneous parameters and characterize its properties by using the Izergin-Korepin analysis. We then show that the product of factors and certain symmetric functions satisfies all the required properties the projected wavefunctions must satisfy. The result is a generalization of [9, 22] and hence can be viewed as a generalization of the Tokuyama for the factorial Schur functions. The Izergin-Korepin analysis views the partition functions as functions of inhomogeneous parameters in the quantum spaces, whereas the arguments initiated in [9] view the partition functions as functions of the free parameter in the auxiliary spaces. The comparison of the two different ways of arguments seems to be interesting.We will use the results of the projected wavefunctions to the algebraic combinatorial study of the generalized Schur functions [34]. For example, two ways of evaluations of the same partition functions can lead to integrable model constructions of algebraic identities of the symmetric functions. For example, two ways of evaluations of the domain wall boundary partition functions, a direct evaluation and an indirect evaluation using the completeness relation and the projected wavefunctions, can give rise to the dual Cauchy formula of the generalized Schur functions. This idea can also be applied to partition functions of integrable models under reflecting boundary to give dual Cauchy identities of the generalized symplectic Schur functions. Further detailed Izergin-Korepin analysis on the domain wall boundary partition functions and the dual projected wavefunctions are required for the studies.There are also studies on deriving Cauchy identities using the domain wall boundary partition functions like an intertwiner, invented in [35]. Deriving algebraic combinatorial properties of symmetric functions using their integrable model realizations is an active line of research. See [36–40] for more examples on Cauchy-type identities and more recent studies on the Littlewood-Richardson coefficients by [33, 41].In any case, in order to conduct these studies, we first of all have to find out what are the explicit functions representing the projected wavefunctions. We think the Izergin-Korepin analysis presented in this paper is a fairly simple way to find out the explicit forms.This paper is organized as follows. In the next section, we first list the generalizedL-operator and introduce the projected wavefunctions. In Section 3, we make the Izergin-Korepin analysis and list the properties needed to determine the explicit form of the projected wavefunctions. In Section 4, we show that the product of factors and certain symmetric functions satisfies all the required properties extracted from the Izergin-Korepin analysis, which means that the product is the explicit form of the projected wavefunctions. Section 5 is devoted to the conclusion of this paper.


## 2. The Generalized Free-Fermion Model and the Projected Wavefunctions

The most fundamental objects in integrable lattice models are theR-matrices and L-operators. The R-matrix of the free-fermion model we treat in this paper is given by(1)Rabz=1+tz0000t1-zt+100t+1zz-10000z+t,acting on the tensor product Wa⊗Wb of the complex two-dimensional space Wa.TheL-operator of the free-fermion model we use as bulk pieces of the projected wavefunctions in this paper is given by(2)Lajz,wj,αj,γj=wj-γjz0000twj+γjzwj00t+1zαjwj+1-αjγjz0000-tαjwj+1-αjγjz,acting on the tensor product Wa⊗Fj of the space Wa and the two-dimensional Fock space at the jth site Fj.The parameterswj, αj, and γj can be regarded as parameters associated with the quantum space Fj. The L-operators giving the Schur functions [9] and factorial Schur functions [22] are a special limit of the generalized L-operator (2) given by(3)Lajz,1,0,0=10000t100t+1zz0000z,(4)Lajz,1,αj,0=10000t100t+1zαj+z0000-tαj+z,respectively.TheL-operator (2) together with the R-matrix (1) satisfies the RLL relation:(5)Rabz1z2Lajz1,wj,αj,γjLbjz2,wj,αj,γj=Lbjz2,wj,αj,γjLajz1,wj,αj,γjRabz1z2,acting on Wa⊗Wb⊗Fj.Let us denote the orthonormal basis ofWa and its dual as {0a,1a} and {〈0|a,〈1|a} and the orthonormal basis of Fj and its dual as {|0〉j,|1〉j} and {j〈0|,j〈1|}. The matrix elements of the L-operator can be written as a〈γ|j〈δ|Laj(z,wj,αj,γj)|α〉a|β〉j, which we will use this form in the next section. See Figure 1 for a pictorial description of the L-operator (2).Figure 1
TheL-operator Laj(z,wj,αj,γj) (2). The horizontal line is the space Wa, and the vertical line is the space Fj.TheR-matrices and the L-operators have origins in statistical physics, and |0〉 or its dual 〈0| can be regarded as a hole state, while |1〉 or its dual 〈1| can be interpreted as a particle state from the point of view of statistical physics. We sometimes use the terms hole states and particle states to describe states constructed from |0〉, 〈0|, |1〉, and 〈1|, since they are convenient for the description of the states. In the quantum inverse scattering method, the Fock spaces Wa and Fj are usually called the auxiliary and quantum spaces, respectively.For later convenience, we also define the following Pauli spin operatorsσ+ and σ- as operators acting on the (dual) orthonormal basis as(6)σ+1=0,σ+0=0,0σ+=1,1σ+=0,σ-0=1,σ-1=0,1σ-=0,0σ-=0.To construct projected wavefunctions, we introduce the monodromy matrixTa(z∣w1,…,wM) (Figure 2(a)) from the generalized L-operator (2) as(7)Taz∣w1,…,wM=LaMz,wM,αM,γM⋯La1z,w1,α1,γ1=Az∣w1,…,wMBz∣w1,…,wMCz∣w1,…,wMDz∣w1,…,wMa∈EndWa⊗F1⊗⋯⊗FM.Figure 2
The monodromy matrixTa(z∣w1,…,wM) (7) (a) and the B-operator B(z∣w1,…,wM) (b).
(a)
(b)The matrix elementsA(z∣w1,…,wM), B(z∣w1,…,wM), C(z∣w1,…,wM), and D(z∣w1,…,wM) are called the ABCD operators, which are 2M×2M matrices acting on the tensor product of the quantum spaces F1⊗⋯⊗FM.To create projected wavefunctions, what is important is theB-operator B(z∣w1,…,wM) (Figure 2(b)) which has the role of creating particles in the quantum spaces F1⊗⋯⊗FM. We next introduce the following state vector.|
Φ
M
,
N
(
z
1
,
…
,
z
N
∣
w
1
,
…
,
w
M
)
〉
∈
F
1
⊗
⋯
⊗
F
M uses the B-operators as(8)ΦM,Nz1,…,zN∣w1,…,wM=Bz1∣w1,…,wM⋯BzN∣w1,…,wMΩM,where |Ω〉M≔|0〉1⊗⋯⊗|0〉M∈F1⊗⋯⊗FM is the vacuum state in the tensor product of quantum spaces.Due to the so-called ice rule of theL-operator a〈γ|j〈δ|Laj(z,wj,αj,γj)|α〉a|β〉j=0 unless α+β=γ+δ, each B-operator creates one particle in the quantum spaces. From this fact and since the state vector (8) is constructed from N-layers of the B-operators acting on the vacuum state |Ω〉M, the state vector (8) is an N-particle state for N≤M. To construct a nonvanishing inner product, we introduce the dual N-particle state:(9)x1⋯xN=01⊗⋯⊗0M∏j=1Nσxj+∈F1∗⊗⋯⊗FM∗,which are states labelling the configurations of particles 1≤x1<x2<⋯<xN≤M.The projected wavefunctionsWM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is defined as the inner product between the state vector (off-shell Bethe vector) |ΦM,N(z1,…,zN∣w1,…,wM)〉 and the N-particle state 〈x1⋯xN|:(10)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=x1⋯xN∣ΦM,Nz1,…,zN∣w1,…,wM.See Figure 3 for a pictorial description of (10).Figure 3
The projected wavefunctionsWM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) (10). This figure illustrates the cases M=5, N=3, x1=2, x2=3, x3=5.In the next section, we examine the properties of the projected wavefunctions. Here we just remark that the projected wavefunctions of the free-fermion model treated in this paper is not symmetric with respect to the spectral parameters{z1,…,zN}. This is in contrast to the case of the projected wavefunctions of the Uq(sl2) six-vertex model, where they are symmetric with respect to the spectral variables, and in that case the Grothendieck polynomials and its quantum group deformation appear. This fact for the properties of the spectral variables of the free-fermion model leads to the Tokuyama formula [10] for the Schur functions, as was first found in [9].


## 3. Izergin-Korepin Analysis

By the Izergin-Korepin analysis, we examine the properties of the projected wavefunctionsWM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) in this section.Proposition 1.
The projected wavefunctionsWM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) satisfy the following properties.
(1)WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is a polynomial of degree N in wM.
(2) The projected wavefunctionsWM,N(zσ(1),…,zσ(N)∣w1,…,wM∣x1,…,xN) with the ordering of the spectral parameters permuted zσ(1),…,zσ(N), σ∈SN are related to the unpermuted one WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) by the following relation:(11)∏1≤j<k≤Nσj>σkzσj+tzσkWM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏1≤j<k≤Nσj>σkzσk+tzσjWM,Nzσ1,…,zσN∣w1,…,wM∣x1,…,xN.(3) The following recursive relations between the projected wavefunctions hold if xN=M:(12)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=γMzN=γMNzN∏j=1N-1zj+tzN∏j=1M-11-αjγjzN+αjwj×WM-1,N-1z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1.When evaluated at wM=0, we have(13)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=0=0.
IfxN≠M, the following factorizations hold for the projected wavefunctions:(14)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏j=1NwM-γMzjWM-1,Nz1,…,zN∣w1,…,wM-1∣x1,…,xN.(4) The following holds for the case N=1, xN=M:(15)WM,1z∣w1,…,wM∣M=wM∏k=1M-11-αkγkz+αkwk.Proof.
Let us first show Properties (1) and (3) for the casexN=M.
To show Property (1) whenxN=M, we first express the projected wavefunctions in terms of the vertical transfer matrix:(16)TjNwj;z1,…,zN=La1jz1,wj,αj,γj⋯LaNjzN,wj,αj,γj∈EndWa1⊗⋯⊗WaN⊗Fj.Using this vertical transfer matrix, the projected wavefunctions can be rewritten as(17)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN-1,M=0⊗N1Mx1⋯xN-1TMNwM;z1,…,zN⋯T1Nw1;z1,…,zN1⊗NΩM,(18)0⊗N=0a1⊗⋯⊗0aN,(19)1⊗N=1a1⊗⋯⊗1aN.
Inserting the completeness relation in one particle sector(20)∑j=1N0j-1,1,0N-j0j-1,1,0N-j=Id,0j-1,1,0N-j=0a1⊗⋯⊗0aj-1⊗1aj⊗0aj+1⊗⋯⊗0aN,0j-1,1,0N-j=0a1⊗⋯⊗0aj-1⊗1aj⊗0aj+1⊗⋯⊗0aN,into (19), we have(21)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN-1,M=∑j=1N0⊗N1MTMNwM;z1,…,zN0j-1,1,0N-j0M×x1⋯xN-10j-1,1,0N-jTM-1NwM-1;z1,…,zN⋯T1Nw1;z1,…,zNΩM-11⊗N.In the right-hand side of (21), the parameter wM depends only on 〈0|⊗NM〈1|TMN(wM;z1,…,zN)|0j-1,1,0N-j〉|0〉M, whose matrix elements can be easily calculated from its graphical representation as(22)0⊗NM1TMNwM;z1,…,zN0j-1,1,0N-j0M=wM∏k=1j-1twM+γMzk∏k=j+1NwM-γMzk.Since the matrix elements (22) are a polynomial of degree N in wM, one finds that the projected wavefunctions are a polynomial of degree N in wM.
Let us next show Property (3) for the casexN=M. We first remark that since the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) are a polynomial of degree N in wM, one needs to evaluate N+1 distinct points in wM for the Izergin-Korepin trick to be successful. Equation (12) is the result of the evaluation at the point wM=γMzN. The (N-1) points wM=γMzj, j=1,…,N-1 can be evaluated using Property (2); hence if one shows that certain functions satisfy Property (2), it remains to consider the evaluation at wM=γMzN. The evaluation at wM=γMzN essentially gives evaluations at N distinct points. We need one more point to be evaluated. An easy point to be evaluated is wM=0, whose result is (13). Let us show these two results of the evaluations.
The recursion relation (12) can be shown as follows. First, from the decomposition (21) and the explicit form of the matrix elements (22), one finds that, after the substitution wM=γMzN, only the term j=N of the sum in (21) survives and we have(23)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN-1,MwM=γMzN=γMNzN∏j=1N-1zj+tzN×x1⋯xN-10N-1,1TM-1NwM-1;z1,…,zN⋯T1Nw1;z1,…,zNΩM-11⊗N.Since we can calculate the right-hand side of (23) furthermore as(24)x1⋯xN-10N-1,1TM-1NwM-1;z1,…,zN⋯T1Nw1;z1,…,zNΩM-11⊗N=x1⋯xN-10N-1TM-1N-1wM-1;z1,…,zN-1⋯T1N-1w1;z1,…,zN-1ΩM-11⊗N-1×aN1M-1ΩLaN,M-1zN,wM-1,αM-1,γM-1⋯LaN,1zN,w1,α1,γ11aNΩM-1=WM-1,N-1z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1∏j=1M-11-αjγjzN+αjwj,we can express the evaluation of WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) at wM=γMzN as(25)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=γMzN=γMNzN∏j=1N-1zj+tzN∏j=1M-11-αjγjzN+αjwj×WM-1,N-1z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1.
The evaluation atwM=0 (13) can be easily seen by the expansion (21) and the fact that all the matrix elements (22) contain the factor wM.
Properties (1) and (3) for the casexN≠M can be shown much easier. Using the ice rule a〈γ|j〈δ|Laj(z,wj,αj,γj)|α〉a|β〉j=0, one can easily find the following factorization:(26)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=0⊗NM0x1⋯xNTMNwM;z1,…,zN⋯T1Nw1;z1,…,zN1⊗NΩM=0⊗NM0TMNwM;z1,…,zN0⊗N0M×0⊗Nx1⋯xNTM-1NwM-1;z1,…,zN⋯T1Nw1;z1,…,zN1⊗NΩM-1=∏j=1Naj0M0Laj,Mzj,wM,αM,γM0aj0M×WM-1,Nz1,…,zN∣w1,…,wM-1∣x1,…,xN=∏j=1NwM-γMzjWM-1,Nz1,…,zN∣w1,…,wM-1∣x1,…,xN.This shows Properties (1) and (3) for the case xN≠M.
Property (2) can be shown as follows. Using theRLL relation repeatedly, one gets the intertwining relation between the monodromy matrices:(27)Rabz1z2Taz1∣w1,…,wMTbz2∣w1,…,wM=Tbz2∣w1,…,wMTaz1∣w1,…,wMRabz1z2.An element of the intertwining relation (27) gives the commutation relation between the B-operators:(28)z2+tz1Bz1∣w1,…,wMBz2∣w1,…,wM=Bz2∣w1,…,wMBz1∣w1,…,wMz1+tz2.Since the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) (10) are constructed from N-layers of B-operators, the effect of reordering the spectral parameters of the B-operators can be traced using the commutation relation (28).
What finally remains is Property (4), which can also easily calculated.Before presenting the solution in the next section, we explain here why the Izergin-Korepin analysis uniquely defines the projected wavefunctions. The idea is based on the following fact: if there are two polynomialsf(w) and g(w) of w of degree N, and the evaluations of the two polynomials at N+1 distinct points are the same (f(w)=g(w) for w=zj, j=1,…,N+1 such that zj≠zk, j≠k), then the two polynomials are exactly the same, that is, f(w)=g(w) for all w. The idea of Izergin-Korepin analysis is to relate the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) to smaller ones by using the above fact. The point is to regard WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) as a polynomial of a single variable wM. By Property (1) in Proposition 1, WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is a polynomial of degree N in wM. If xN=M, one can evaluate the projected wavefunction at the following N+1 points: wM=γMzj, j=1,…,N, wM=0. The evaluations at wM=γMzN and wM=0 can be obtained by its graphical representation and can be expressed by using the smaller projected wavefunction WM-1,N-1(z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1), which is (12) and (13) of Property (3). The evaluations at (N-1) points wM=γMzj, j=1,…,N-1 can be obtained from the evaluation at wM=γMzN using Property (2). This idea is essentially the same with the Izergin-Korepin analysis for the domain wall boundary partition functions.For the case of projected wavefunctions, there is another case we have to consider: the case whenxN≠M. For this case, it is easier to connect the projected wavefunctions from its graphical description, and we have (14). Note that the smaller projected wavefunctions connected are WM-1,N(z1,…,zN∣w1,…,wM-1∣x1,…,xN) which are different from the one for the case when xN=M.In both casesxN=M and xN≠M, we are able to connect the projected wavefunctions of different sizes, and, continuing this process successively, the relations can be regarded as recursion relations between projected wavefunctions. For the Izergin-Korepin analysis to be successful such that it gives the uniqueness of the projected wavefunctions, we need the initial condition for the recursion relations, and it is Property (4) in Proposition 1. Hence, if one finds an explicit function satisfying all the properties in Proposition 1, it is the one representing the projected wavefunctions. This is given in the next section.


## 4. Generalized Schur Functions

Definition 2.
We define the following symmetric functionFM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) which depends on the symmetric variables z1,…,zN, complex parameters w1,…,wM, α1,…,αM, γ1,…,γM and integers x1,…,xN satisfying 1≤x1<⋯<xN≤M:(29)FM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏j=1Nwxj1∏1≤j<k≤Nzk-zj∑σ∈SN-1σ∏j=1N∏k=xj+1Mwk-γkzσj×∏j=1N∏k=1xj-11-αkγkzσj+αkwk.The symmetric functionFM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) (29) is a generalization of the (factorial) Schur functions. Equation (29) can be rewritten in the form using Young diagrams as(30)FM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=Fλ+δz∣w1,…,wM∏1≤j<k≤Nzj-zk.Here, z={z1,…,zN} is a set of variables and λ denotes a Young diagram λ=(λ1,λ2,…,λN) with weakly decreasing nonnegative integers λ1≥λ2≥⋯≥λN≥0, and δ=(N-1,N-2,…,0). Fμ(z∣{α}∣{γ}) is an N×N determinant:(31)Fμz∣w1,…,wM=detNfμjzk∣w1,…,wM,where(32)fμz∣w1,…,wM=wμ+1∏j=1μ1-αjγjz+αjwj∏j=μ+2Mwj-γjz.The Young diagrams λ=(λ1,λ2,…,λN)∈ZN (M-N≥λ1≥λ2≥⋯≥λN≥0) in form (30) and the sequence of integers x1,…,xN satisfying 1≤x1<⋯<xN≤M in (29) representing FM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) are connected by the translation rule λj=xN-j+1-N+j-1, j=1,…,N.In the limitwj=1, j=1,…,M, γj=0, j=1,…,M, we can see from form (30) that FM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) reduces to the factorial Schur functions. If one furthermore sets αj=0, j=1,…,M, it further reduces to the Schur functions.We have the following correspondence between the projected wavefunctions of the integrable model and the generalized Schur functionFM,N(z1,…,zN∣w1,…,wM∣x1,…,xN).Theorem 3.
The projected wavefunctions of the generalized free-fermion modelWM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) are explicitly expressed as the product of factors ∏1≤j<k≤N(zj+tzk) and the symmetric function FM,N(z1,…,zN∣w1,…,wM∣x1,…,xN):(33)WM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏1≤j<k≤Nzj+tzkFM,Nz1,…,zN∣w1,…,wM∣x1,…,xN.In the limitswj=1, j=1,…,M, γj=0, j=1,…,M, (33) reduces to the main theorem of Bump-McNamara-Nakasuji [22]. Taking the limit αj=0, j=1,…,M furthermore, one gets the main theorem of Bump-Brubaker-Friedberg [9].Proof.
Let us denote the right-hand side of (33) as GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN):(34)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xN≔∏1≤j<k≤Nzj+tzkFM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏j=1Nwxj∏1≤j<k≤Nzj+tzk∏1≤j<k≤Nzk-zj∑σ∈SN-1σ∏j=1N∏k=xj+1Mwk-γkzσj×∏j=1N∏k=1xj-11-αkγkzσj+αkwk.
We prove this theorem by showing thatGM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) satisfies all four properties in Proposition 1.
To show Property (1), first note that the factor∏j=1N∏k=xj+1M(wk-γkzσ(j)) in GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is a polynomial of degree N-1 in wM if xN=M and degree N if xN≠M. For the case xN≠M, one sees that the dependence on wN just only comes from this factor. For the case xN=M, there is a factor wM coming from the overall factor ∏j=1Nwxj. Thus, GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is a polynomial of degree N in wM for both cases xN=M and xN≠M; hence Property (1) is proved.
Let us next show Property (2). First, note thatFM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) which is a part of GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) that is symmetric with respect to z1,…,zN since both the denominator ∏1≤j<k≤N(zk-zj) and the numerator(35)∏j=1Nwxj∑σ∈SN-1σ∏j=1N∏k=xj+1Mwk-γkzσj∏j=1N∏k=1xj-11-αkγkzσj+αkwk,are antisymmetric with respect to simple permutations of the spectral parameters z1,…,zN. This means(36)FM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=FM,Nzσ1,…,zσN∣w1,…,wM∣x1,…,xN.Looking at the other factor ∏1≤j<k≤N(zj+tzk) which constructs the function GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN), we have(37)∏1≤j<k≤Nσj>σkzσj+tzσk∏1≤j<k≤Nzj+tzk=∏1≤j<k≤Nσj>σkzσk+tzσj∏1≤j<k≤Nzσj+tzσk.From (36), (37), and the fact that GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) is defined as a product of ∏1≤j<k≤N(zj+tzk) and FM,N(z1,…,zN∣w1,…,wM∣x1,…,xN), we have(38)∏1≤j<k≤Nσj>σkzσj+tzσkGM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏1≤j<k≤Nσj>σkzσk+tzσjGM,Nzσ1,…,zσN∣w1,…,wM∣x1,…,xN.We have shown that GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) satisfies the same relation the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) must satisfy. Hence Property (2) is proved.
Next we show Property (3). We first prove that the functionGM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) satisfies (12) and (13) for the case xN=M. To prove (12), we first note that the factor(39)∏j=1N∏k=xj+1Mwk-γkzσj,in each summand essentially becomes(40)∏j=1N-1∏k=xj+1Mwk-γkzσj.Concentrating on the factor ∏j=1N-1(wM-γMzσ(j)) from (40), one finds this factor vanishes unless σ satisfies σ(N)=N if one substitutes wM=γMzN.
Therefore, only the summands satisfyingσ(N)=N in (29) survive after the substitution wM=γMzN. Keeping this in mind, one rewrites GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) evaluated at wM=γMzN by using the symmetric group SN-1 where every σ′∈SN-1 satisfies {σ′(1),…,σ′(N-1)}={1,…,N-1} as follows:(41)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=γMzN=γMzN∏j=1N-1wxj∏1≤j<k≤N-1zj+tzk∏j=1N-1zj+tzN∏1≤j<k≤N-1zk-zj∏j=1N-1zN-zj×∑σ′∈SN-1-1σ′∏j=1N-1∏k=xj+1M-1wk-γkzσ′j∏j=1N-1γMzN-zσ′j×∏j=1N-1∏k=1xj-11-αkγkzσ′j+αkwk∏k=1M-11-αkγkzN+αkwk.One easily notes that the factors ∏k=1M-1{(1-αkγk)zN+αkwk} in the sum are independent of the permutation SN-1′. One also finds that the product of factors 1/∏j=1N-1(zN-zj) and ∏j=1N-1γM(zN-zσ′(j)) can be simplified as(42)1∏j=1N-1zN-zj∏j=1N-1γMzN-zσ′j=1∏j=1N-1zN-zj∏j=1N-1γMzN-zj=γMN-1.
Thus, (41) can be rewritten furthermore as(43)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=γMzN=γMNzN∏j=1N-1zj+tzN∏j=1M-11-αjγjzN+αjwj×∏j=1N-1wxj∏1≤j<k≤N-1zj+tzk∏1≤j<k≤N-1zk-zj∑σ′∈SN-1-1σ′∏j=1N-1∏k=xj+1M-1wk-γkzσ′j×∏j=1N-1∏k=1xj-11-αkγkzσ′j+αkwk.Since(44)∏j=1N-1wxj∏1≤j<k≤N-1zj+tzk∏1≤j<k≤N-1zk-zj∑σ′∈SN-1-1σ′∏j=1N-1∏k=xj+1M-1wk-γkzσ′j×∏j=1N-1∏k=1xj-11-αkγkzσ′j+αkwk=GM-1,N-1z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1,one finds that (43) is nothing but the following recursion relation for the function GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN):(45)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xNwM=γMzN=γMNzN∏j=1N-1zj+tzN∏j=1M-11-αjγjzN+αjwj×GM-1,N-1z1,…,zN-1∣w1,…,wM-1∣x1,…,xN-1,which is exactly the same recursion relation the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) must satisfy; hence (12) is shown. Equation (13) can be shown immediately since GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) evaluated at wM=0 becomes zero due to the overall factor ∏j=1Nwxj in FM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) and the fact that we are dealing the case xN=M.
Now let us examine the casexN≠M. This can be shown in a similar but much simpler way. We rewrite GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) as(46)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏j=1Nwxj∏1≤j<k≤Nzj+tzk∏1≤j<k≤Nzk-zj∑σ∈SN-1σ∏j=1N∏k=xj+1M-1wk-γkzσj∏j=1NwM-γMzσj×∏j=1N∏k=1xj-11-αkγkzσj+αkwk.Noting that(47)∏j=1NwM-γMzσj=∏j=1NwM-γMzj,we can take this factor out of the sum in (46) and we get(48)GM,Nz1,…,zN∣w1,…,wM∣x1,…,xN=∏j=1NwM-γMzj∏j=1Nwxj∏1≤j<k≤Nzj+tzk∏1≤j<k≤Nzk-zj∑σ∈SN-1σ∏j=1N∏k=xj+1M-1wk-γkzσj×∏j=1N∏k=1xj-11-αkγkzσj+αkwk=∏j=1NwM-γMzjGM-1,Nz1,…,zN∣w1,…,wM-1∣x1,…,xN, which is also exactly the recursion relation the projected wavefunctions GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) must satisfy for the case xN≠M.
Finally it is trivial to check from its definition that(49)GM,1z∣w1,…,wM∣M=wM∏k=1M-11-αkγkz+αkwk;hence Property (4) is shown.
Since we have proved that the functionGM,N(z1,…,zN∣w1,…,wM∣x1,…,xN) satisfies all Properties (1), (2), (3), and (4) in Proposition 1, we conclude that it is the explicit form of the projected wavefunctions WM,N(z1,…,zN∣w1,…,wM∣x1,…,xN)=GM,N(z1,…,zN∣w1,…,wM∣x1,…,xN).


## 5. Conclusion

In this paper, we studied the generalized free-fermion model in an external field. We applied the Izergin-Korepin analysis on the projected wavefunctions which is a generalization of the Izergin-Korepin analysis on the domain wall boundary partition functions, which was recently done for the case of theUq(sl2) six-vertex model in [30]. We extracted the properties about the degree, symmetry, recursion relations, and initial conditions the projected wavefunctions satisfy. Next we proved that the product of factors and certain symmetric functions satisfies all the required properties; hence it represents the projected wavefunctions. The result can be regarded as an extension of the Tokuyama formula for the (factorial) Schur functions by Bump-Brubaker-Friedberg [9] and Bump-McNamara-Nakasuji [22].The result obtained in this paper can also be proved by using the arguments initiated in [9], which views the partition functions as functions of the free parameter in the auxiliary spaces. The Izergin-Korepin analysis used in this paper views the partition functions as functions of inhomogeneous parameters in the quantum spaces. The comparison of the two different ways of arguments seems to be interesting. We use the result obtained in this paper to study algebraic combinatorial properties of the generalized Schur functions in our forthcoming paper [34]. Extending the Izergin-Korepin analysis to other boundary conditions of the generalized free-fermion model is one of the interesting topics regarding this paper. There may be some cases in which the Izergin-Korepin analysis is suitable and some other cases in which the arguments in [9] are useful. We think that developing various techniques are useful for the study of partition functions of integrable lattice models.



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