数理解析研究所講究録
第 810 巻 1992 年 22-38

22

COMPLETELY Z SYMMETRIC$R$

MATRIX

早大・理工 上野喜三雄 (Kimio UENO)

早大・理工 澁川陽一 (Youichi SHIBUKAWA)

1. INTRODUCTION
In this paper, we shall introduce an infinite-dimensional

limiting case $narrow\infty$ of the completely
as the$R$$\mathbb{Z}_{n}$$R$matrix related to thesymmetric $R$ matrix. This is not the same

matrix of Gaudin [8], $G6mez$ -Sierra [9], and Fateev-Zamolodchikov [5]. Of
matrix satisfies the Yang-Baxter equation

course, this

$R$

(1.1)

$R_{12}(\lambda_{1})R_{13}(\lambda_{1}+\lambda_{2})R_{23}(\lambda_{2})=R_{23}(\lambda_{2})R_{13}(\lambda_{1}+\lambda_{2})R_{12}(\lambda_{1})$By means of the Fourier transformation, we shall give an

This

$R$

$R$

operator on.
$C^{\infty}(S^{1}\cross S^{1})$.

operator is also a solution of the Yang-Baxter equation (1.1). Moreover weshall apply the fusion procedure to thedimensional$R$

matrix from the

$R$

$R$

operator, and shall construct a finite-

operator.

1991 Mathematics Subject Classification. Primary $81R50,82B23$ ; Secondary $17B37,82B20$ .

23The reason why we begin investigations of such an infinite-dimensional$R$

matrixis as follows.

The quantum group is useful for the geometric interpretation of Macdonald’s sym-

metric polynomials. In fact, Ueno-Takebayashi [17] and Noumi [13] proved that Mac-

donald’s symmetric polynomials are the zonal spherical functions on some quantum
homogeneous space. Although Macdonald’s symmetric polynomials have two parameters, they dealed with the case of only one parameter. Further it is rather morenatural to consider Macdonald’s symmetric functions than Macdonald’s symmetricpolynomials. Roughly speaking, Macdonald’s symmetric functions are the polynomials with infinite variables. Then we aim at defining the quantum groupa deformation ofwith two parameters$U(gl_{\infty})$

$q$

and

$t$

$U_{q,t}(gl_{\infty})$,for giving the geometric in-terpretation of Macdonald’s symmetric functions. In view of Sklyanin algebra [16],it is important to find out the infinite-dimensional
quantum group$U_{q,t}(gl_{\infty})$$R$matrix which can define the.

On the other hand, Freund-Zabrodin [7] and Zabrodin [19] showed that Macdonald’s symmetric functions are related to the limiting case $narrow\infty$ of the completely$\mathbb{Z}_{n}$symmetric model.

The details will be discussed in a forthcoming paper [15].

24
2. COMPLETELY$\mathbb{Z}_{n}$Let us quickly review the completely

SYMMETRIC $R$ MATRIX$\mathbb{Z}_{n}$symmetric$R$matrix. We denote by

the Boltzmann weight for a single vertex with bond states $i,j,$$R_{ij}^{k\ell}(\lambda)$

These Boltzmann weights$R_{ij}^{k\ell}(\lambda)$

define a matrix$R(\lambda)$

in the standard basis. This matrix$R(\lambda)$

is said to be completely$\{e_{i}\otimes e_{j};i,j\in \mathbb{Z}_{n}\}$forsymmetric if

satisfies the conditions below:$R_{ij}^{k\ell}(\lambda)$$\mathbb{C}^{n}\otimes \mathbb{C}^{n}$$R_{ij}^{k\ell}(\lambda)=0$(2)$R_{i+p,j+P}^{k+p,\ell+p}(\lambda)=R_{ij}^{k\ell}(\lambda)$$\mathbb{Z}_{n}$.

$\mathbb{Z}_{n}$unless $i+j=k+\ell$ mod ,(1)

Because of$k,l\in \mathbb{Z}_{n}$

$n$

for

$\forall i,j,$symmetry, there exists(2.1)$k,l\in \mathbb{Z}_{n}$.

$S^{ab}(\lambda)(a, b\in \mathbb{Z}_{n})$$R_{ij}^{kl}(\lambda)=\delta_{i+j,k+\ell S^{k-i,\ell-i}(\lambda)}$We define the Jacobi theta functions$\theta\{\begin{array}{l}ab\end{array}\}(z, \tau)$satisfying.

of rational characteristics$a,$ $b \in\frac{1}{n}\mathbb{Z}$

by
(2.2)

$\theta\{\begin{array}{l}ab\end{array}\}(z, \tau)=\sum_{m\in \mathbb{Z}}\exp[\pi\sqrt{-1}(m+a)^{2}\tau+2\pi\sqrt{-1}(m+a)(z+b)]$,

and put(23)

Here $Im\tau>0$ , and$S^{ab}( \lambda)=\frac{\theta[\frac{b-a}{n\frac{1}{2}}+\frac{1}{2}](\lambda+\kappa,n\tau)}{\theta[-\frac{a}{n}\frac{1}{2}+\frac{1}{2}](\kappa,n\tau)\theta[\frac{b}{n}\frac{+1}{2}\frac{1}{2}](\lambda,n\tau)}$$\kappa$

is a constant. These weights

Baxter equation (1.1). This completely$\mathbb{Z}_{n}$$S^{ab}(\lambda)$

symmetric

Belavin [2], Cherednik [3] [4], and Richey-Tracy [14].

$R$.

give a solution of the Yangmatrix has been studied by

25
The weight$S^{ab}(\lambda)$

is expressed as follows.

(2.4)$S^{ab}(\lambda)=q^{-\frac{n}{8}}\tilde{S}^{ab}(\lambda)$,$\tilde{S}^{ab}(\lambda)$$=q^{-\frac{ab}{n}}e^{2\pi\sqrt{-1}(\frac{b}{n}\kappa-\frac{a}{n}\lambda)}$(25)$\cross\frac{\Pi_{m=1}^{\infty}(1-q^{n\langle m-1)-(b-a)}e^{-2\pi\sqrt{-1}(\lambda+\kappa)})(1-q^{nm+\langle b-a)}e^{2\pi\sqrt{-1}(\lambda+\kappa)})}{\Pi_{m=1}^{\infty}(1-q^{n\langle m-1)+a}e^{-2\pi\sqrt{-1}\kappa})(1-q^{nm-a)}e^{2\pi\sqrt{-1}\kappa})}$$\cross\frac{(1-q^{nm+b}e^{2\pi\sqrt{-1}\lambda})}{(1-q^{n\langle m-1)-b}e^{-2\pi\sqrt{-1}\lambda})}$Here$q=e^{2\pi\sqrt{-1}\tau}$

(2.6)

.

. Taking $Im\tau>0$ into account,$\lim_{narrow\infty}\tilde{S}^{ab}(\lambda)=\frac{1-q^{a-b}e^{-2\pi\sqrt{-1}(\lambda+\kappa)}}{(1-q^{a}e^{-2\pi\sqrt{-1}\kappa})(1-q^{-b}e^{-2\pi\sqrt{-1}\lambda})}$3. COMPLETELY$\mathbb{Z}$SYMMETRIC $R$ MATRIXLet us consider an infinite-dimensional
(3.1)

Here

is the matrix unit incompletely$\mathbb{Z}$$\mathbb{Z}\cross \mathbb{Z}$symmetry on such an$S^{ab}(\lambda)(a, b\in \mathbb{Z})$

(3.2)

$R$

matrix,$R( \lambda)=\sum_{i,j,k,f\in \mathbb{Z}}R_{i,j^{f}}^{k}(\lambda)E_{ik}\otimes E_{j\ell}$$E_{ik}$.

.

matrix algebra. We impose a constraint of the$R$matrix. Namely, we assume that there exists

such that$R_{i,j}^{k,l}(\lambda)=\delta_{i+j,k+f}S^{k-i,l-i}(\lambda)$.

26
Then the Yang-Baxter equation (1.1) reads, in terms of these weights$S^{ab}(\lambda)$

asfollows:$\sum_{k=-\infty}^{\infty}S^{k-i_{1},i_{2}-k}(\lambda_{1})S^{j_{1}-k,i_{3}-j_{1}}(\lambda_{1}+\lambda_{2})S^{j_{2}-i_{1}-i_{2}+k,j_{3}-i_{1}-i_{2}+k}(\lambda_{2})$$(3.3)$$= \sum_{k=-\infty}^{\infty}S^{j_{1}+j_{2}-i_{2}-k,j_{3}-i_{1}-i_{2}+k}(\lambda_{2})S^{k-i_{1},j_{3}-i_{1}}(\lambda_{1}+\lambda_{2})S^{j_{1}-k,j_{2}-k}(\lambda_{1})$for all$i_{1},$$i_{2},$$i_{3},j_{1},j_{2},j_{3}\in \mathbb{Z}s.t$,. $i_{1}+i_{2}+i_{3}=j_{1}+j_{2}+j_{3}$ .

In view of the limiting case $narrow\infty(2.6)$ of the completely$Z_{n}$

symmetric$R$matrix,

we find a solution of the Yang-Baxter equation.

Theorem 3.1. We assume $|q|<1$ , and then
(3.4)

$S^{ab}( \lambda)=-\frac{q^{a}}{e^{2\pi\sqrt{-1}\kappa}-q^{a}}+\frac{q^{b}}{e^{2\pi\sqrt{-1}\lambda}-q^{b}}$is a solution of the equation (3.3). Here$\kappa$

is an arbitrary constant.

Note that the both sides of equation (3.3) are absolutely convergent. We can prove

the theorem above by residue calculus in a variable$u_{1}def=e^{2\pi\sqrt{-1}\lambda_{1}}$Remark 3.1. We put $q=e^{2\pi\sqrt{-1}\tau}(Im\tau>0)$ , and replace$\kappa$

withlimiting case $qarrow 1$ ,(3.5)$(q-1)S^{ab}( \tau\lambda)|_{qarrow 1}=-\frac{1}{\kappa-a}+\frac{1}{\lambda-b}$

It is same as the weight$S^{ab}$given by Gaudin [8].

.

.

$\tau\kappa$in (3.4). In the

274.
We shall realize the

$R$

$R$

OPERATOR

matrix as an operator on some function space making useof the formula below (see [11] p.446):$\frac{1}{\pi}\frac{\theta_{1}’(0)\theta_{1}(x+y)}{\theta_{1}(x)\theta_{1}(y)}$(4.1)$= \cot\pi x+\cot\pi y+4\sum_{m,n=1}^{\infty}q^{mn}\sin 2\pi(mx+ny)$$|Imx|<Im\tau,$ $|Imy|<Im\tau$ .

Here$\theta_{1}(z)$is the elliptic theta function,(4.2)$\theta_{1}(z)=2q^{\frac{1}{8}}\sin\pi z\prod_{m=1}^{\infty}(1-q^{m})(1-2q^{m}\cos 2\pi z+q^{2m})$.

Using the formula above, we can compute the Fourier transformation of the Boltz-

mann weight$R_{i,j}^{k,l}$(cf. Gaudin [8]).

Theorem 4.1. For$x,$ $y\in \mathbb{R},$$|Im\lambda|<Im\tau$, and $|Im\kappa|<Im\tau$ ,$\sum_{i,j\in \mathbb{Z}}R_{i,j}^{k,\ell}(\lambda)e^{2\pi\sqrt{-1}(ix+jy)}$(4.3)$=G(x-y:\lambda)e^{2\pi\sqrt{-1}\langle\ell x+ky)}-G(x-y:\kappa)e^{2\pi\sqrt{-1}(kx+\ell y)}$

,

where
(4.4)

Let

$G(x: \lambda)=\frac{1\theta_{1}’(0)\theta_{1}(\lambda+x)}{2\pi\sqrt{-1}\theta_{1}(\lambda)\theta_{1}(x)}$$\mathcal{V}$The setbe the set of$\{\varphi_{k}; k\in \mathbb{Z}\}$$C^{\infty}$-functions on the unit circle

is a basis of$\mathcal{V}$$S^{1}$.

, and let$\varphi_{k}(x)=e^{2\pi\sqrt{-1}kx}$. Then the theorem above says that the$R$.

matrix28
gives rise to a linear operator on$\mathcal{V}\otimes \mathcal{V}$defined by$(R(\lambda)(\varphi_{k}\otimes\varphi_{\ell}))(x, y)$(4.5)$def=G(x-y : \lambda)(\varphi_{k}\otimes\varphi_{\ell})(y,x)-G(x-y :In this case we call$R(\lambda)$

a non-local operator on
(46)

Here

the

$R$

\kappa)(\varphi_{k}\otimes\varphi_{f})(x, y)$operator. Further the $R$ operator can be regarded as$\mathcal{V}\otimes \mathcal{V}\wedge=C^{\infty}(S^{1}\cross S^{1})$,$R(\lambda)=G(\lambda)\sigma-G(\kappa)$$G(\lambda)$

and$\sigma$

is an operator on(4.7)

defined by$(\sigma\varphi)(x,y)=\varphi(y,x)$

We can see that the$\varphi\in \mathcal{V}^{\otimes^{\wedge}2}$$\mathcal{V}^{\otimes^{\wedge}2}$.

$(G(\lambda)\varphi)(x, y)=G(x-y:\lambda)\varphi(x, y)$

(48)

$R$

.

,

.

operator actually belongs to End$(\mathcal{V}^{\otimes^{\wedge}2})$i.e.

$R(\lambda)\varphi\in \mathcal{V}^{\otimes^{\wedge}2}$for.

Now let us establish the Yang-Baxter equation for the
we define the operator(4.9)

where for

$R_{ij}(\lambda)\in End(\mathcal{V}^{\otimes^{\wedge}N})$$R$

operator (4.6). For $N\geq 2$ ,$(1 \leq i,j\leq N, i\neq j)$$R_{ij}(\lambda)=G_{ij}(\lambda)\sigma_{ij}-G_{ij}(\kappa)$.

,$\varphi\in \mathcal{V}^{\otimes^{\wedge}N}$(4.10)$(G_{ij}(\lambda)\varphi)(x_{1}, x_{2}, \ldots, x_{N})=G(x;-x_{j} :(4.11)$(\sigma_{ij}\varphi)(\ldots, x_{i}, \ldots, x_{j}, \ldots)=\varphi(\ldots, x_{j}, \ldots, x_{i}, \ldots)$\lambda)\varphi(x_{1}, x_{2}, \ldots,x_{N})$

.

,

29
Theorem 4.2.

$R(\lambda)$

satisfies the Yang-Baxter equation (1.1) in End$(\mathcal{V}^{\otimes^{\wedge}3})$. Namely

the operators defined by (4.9) for $N=3$ satisfy(4.12)$R_{12}(\lambda_{1})R_{13}(\lambda_{1}+\lambda_{2})R_{23}(\lambda_{2})=R_{23}(\lambda_{2})R_{13}(\lambda_{1}+\lambda_{2})R_{12}(\lambda_{1})$.

This theorem can be verified in a slightly more general context.

Proposition 4.3. If an analytic function$\theta(x)$satisfies the three term equation$\theta(x+y)\theta(x-y)\theta(z+w)\theta(z-w)$$+\theta(x+z)\theta(x-z)\theta(w+y)\theta(w-y)$$(4.13)$$+\theta(x+w)\theta(x-w)\theta(y+z)\theta(y-z)$$=0$ ,

then the operator(4.14)$R(\lambda)^{d}=^{ef}G(\lambda)\sigma-G(\kappa)$

is a solution of the Yang-Baxter equation in the same sense as in Theorem 4.2. Here(4.15)$(G( \lambda)\varphi)(x, y)=\frac{\theta’(0)\theta(\lambda+x-y)}{\theta(\lambda)\theta(x-y)}\varphi(x,y)$

for a function
Remark 4.1.

$\varphi(x, y)$.

(1) The elliptic theta function$\theta_{1}(x)$satisfies (4.13). It is actually

Fay’s trisecant formula (see [6] p.33-35). It is worthwhile noticing that theprime form on an elliptic curve is given by(4.16)$E(x, y) \sqrt{dxdy}=\frac{\theta_{1}(y-x)}{\theta_{1}(0)}$.

30(2) Analytic solutions of the three term equation (4.13) are given by,(4.17)$\theta(x)=\theta_{1}(x)\exp(\frac{1}{2}Ax^{2}+B)$(4.18)$\theta(x)=\sin(\pi x)\exp(\frac{1}{2}Ax^{2}+B)$(4.19)$\theta(x)=x\exp(\frac{1}{2}Ax^{2}+B)$

where

$A$

,

,

and $B$ are arbitrary constants (see [18] p.461). In the situation of

Proposition 4.3, we simply assume
variable and$\mathcal{V}\otimes \mathcal{V}\wedge$$\mathcal{V}$to be a space of functions with one

to be a space of functions with two variables, respectively.

In what follows, we shall assume that the$R$operator is the generalized one in

Proposition 4.3. We state the first inversion relation for$R$.

Proposition 4.4 (The first inversion relation).

(4.20)$R_{12}(\lambda)R_{21}(-\lambda)=\rho(\lambda)id$

,

where

(4.21)$\rho(\lambda)=\{\begin{array}{l}\theta_{1}’(0)^{2}\frac{\theta_{1}(\lambda+\kappa)\theta_{1}(\lambda-\kappa)}{\theta_{1}^{2}(\lambda)\theta_{1}^{2}(\kappa)}\pi^{2}(cot^{2}\pi\kappa-cot^{2}\pi\lambda)\frac{1}{\kappa^{2}}-\frac{1}{\lambda^{2}}\end{array}$$i_{ncase(418)}i_{incase(419)}^{ncase(4...17)},$.

31
5. FUSION PROCEDURE
From the definition of the operator$R(\lambda)$on$V^{\otimes 2}\wedge$, we obtain(5.1)$R(\kappa)=-2G(\kappa)P^{\{-)}$

,

(5.2)

$R(-\kappa)=-2P^{(+)}G(\kappa)$.

Here $P^{(+)}=$ } $(1+\sigma)$ and$P^{\langle-)}= \frac{1}{2}(1-\sigma)$space of symmetric functions and$S^{\langle-)}(V^{\otimes^{\wedge}2})$

are the projectors on$S^{\langle+)}(V^{\otimes^{\wedge}2})$

thethe space of anti-symmetric functions,

respectively. Specializing the spectral parameter in the Yang-Baxter equation (4.12),
we get$R_{13}(\lambda+\kappa)R_{23}(\lambda)P_{12}^{t+)}=P_{12}^{(+)}R_{13}(\lambda+\kappa)R_{23}(\lambda)P_{12}^{(+)}$(5.3)$(\lambda_{1}=\kappa, \lambda_{2}=\lambda)$,$R_{13}(\lambda)R_{12}(\lambda-\kappa)P_{23}^{(+)}=P_{23}^{(+)}R_{13}(\lambda)R_{12}(\lambda-\kappa)P_{23}^{\langle+)}$(5.4)$(\lambda_{1}=\lambda, \lambda_{2}=-\kappa)$.

Taking the equations above into account, we can apply the fusion procedure for vertex

models which was developed in [12] (see also [3] and [10]), to our case.

We define a product of the$\{\varphi(x_{1}, \ldots, x_{L} :

(5.5)

where

y_{1}, \ldots, y_{M})\}$$R$

operators on the function space $V\otimes \mathcal{V}’=$:$\{\begin{array}{l}R_{f\cdot.1’\ldots M’}(\lambda)=R_{f\cdot.M’}(\lambda)R_{f\cdot.M’-1}(\lambda-\kappa)\ldots R_{\ell\cdot.1’}(\lambda-(M-1)\kappa)R_{1\ldots L\cdot.1’\ldots M’}(\lambda)=R_{1\cdot.1’\ldots M},(\lambda+(L-1)\kappa)\ldots R_{L.\cdot 1’\ldots M’}(\lambda)\end{array}$$V’$is a copy of V.

$R_{\ell:k’}(\lambda)$

indicates the action on the variables$x_{\ell}$and$y_{k}$

. Let

32

$S^{\langle+)}(\mathcal{V}^{\otimes^{\wedge}L})$and$S^{\langle-)}(V^{\otimes^{\wedge}L})$be the spaces of symmetric functions and anti-symmetricfunctions, respectively, and let(5.6)

be the projector onto

$P_{1..L}^{(\pm.)}= \sum_{w\in 6_{L}}(\pm 1)^{\ell(w)}w$

Here we denote by$\epsilon_{1},$$P_{1\ldots L}^{\langle\pm)}$$\epsilon_{2}=(+)$(5.7)$\mathfrak{S}_{L}$$S^{(\pm)}(\mathcal{V}^{\otimes^{\wedge}L})$.

the symmetric group and by $l(w)$ the length of, (-), define the operator

$w$

. For

by

$R_{(L):(M)}^{\epsilon_{1}\epsilon_{2}}(\lambda)$$R_{\langle L):\langle M’)}^{e_{1}\epsilon_{2}}(\lambda)=P_{1}^{\epsilon_{1}}\ldots {}_{L}P_{1\ldots M’}^{\epsilon_{2}}R_{1\ldots L:1’\ldots M’}(\lambda)P_{1}^{\epsilon_{1}}\ldots {}_{L}P_{1\ldots M’}^{\epsilon_{2}}$Theorem 5.1. In End,$(S^{\epsilon_{1}}(\mathcal{V}^{\otimes^{\wedge}L})\otimes S^{\epsilon_{2}}(V^{\otimes M})\otimes S^{\epsilon_{3}}(\mathcal{V}’’\otimes^{\wedge}N))\wedge’\wedge\wedge$.

,$R_{(L):\langle M’)}^{\epsilon_{1}\epsilon_{2}}(\lambda_{1})R_{\{L):(N’)}^{\epsilon_{1}e_{3}}(\lambda_{1}+\lambda_{2})R_{(M);(N’)}^{\epsilon_{2}\epsilon_{3}}(\lambda_{2})$$(5.8)$$=R_{\langle M):\{N’)}^{\epsilon_{2}\epsilon_{3}}(\lambda_{2})R_{(L):(N’)}^{\epsilon_{1}\epsilon_{3}}(\lambda_{1}+\lambda_{2})R_{(L):(M’)}^{\epsilon_{1}e_{2}}(\lambda_{1})$where$\epsilon_{1},\epsilon_{2},$$\epsilon_{3}=(+)$,, (-), and $V”$ is a copy of V.

In a forthcoming paper, we will discuss that the equation (5.8) corresponds towhich functional equation of the elliptic theta function.
Example. We give an explicit formula for$R_{1:(M)}^{t+)}$, which is the case $L=1,$

in (5.7).
(5.9)$R_{1:(M’)}^{(+)}( \lambda)=\frac{1}{M!}\sum_{j=1}^{M}\dot{d}_{1’\ldots M’}^{-1}G_{1:\langle M’)}^{t+)}\sigma_{11’}+(-1)^{M}\prod_{j=1}^{M}G_{1j’}(\kappa)$

where(5.10)$(\sigma_{11’}\varphi)(x_{1} :(5.11)$(\sigma_{1’\ldots M’}\varphi)(x_{1} :y_{1}, \ldots,y_{-}M)=\varphi(y_{1} :,x_{1}, y_{2}, \ldots, y_{M})$y_{1}, \ldots, y_{M})=\varphi(x_{1} :y_{2}, \ldots, y_{M}, y_{1})$,,$\epsilon_{2}=(+)$

33
and

is a multiplication operator defined inductively by$G_{1:(M)}^{t+)}$$G_{1:(M’)}^{(+)}(x :y_{1}, \ldots, y_{M} :$= \sum_{j=2}^{M}\{G(x_{1}-y_{j} :\lambda)$\lambda)G_{1:\langle(M-1)’)}^{(+)}(y_{j} :y_{1}, \ldots,\hat{y}_{j}\ldots, y_{M} :\lambda-\kappa)$(5.12)$-G(x_{1}-yj :\kappa)G_{1:(\langle M-1)’)}^{t+)}(x_{1} :$+(-1)^{M-1}(M-1)!G(x_{1}-y_{1} :y_{1}, \ldots,\hat{y}j\cdots, y_{M} :\lambda)\prod_{j=2}^{M}G(y_{1}-y_{j} :\kappa)$6. FINITE-DIMENSIONAL REPRESENTATION OF$R$\lambda-\kappa)\}$.
OPERATORFirst we formulate the notion of finite-dimensional representations of theerator. Let$S$

be a finite index set, and let

subspace of V with a basis

Definition 6.1. If the$R$$\{f_{\alpha} :$R(\lambda)$$R(\lambda)|_{Votimes V\}\in End(V^{\mathfrak{F}}\otimes V^{\mathfrak{F}})$We should remark that

$R^{\iota}?(\lambda)$

op-

be a finite-dimensional.

operator preserves

dimensional representation ofof\alpha\in S\}$

$V^{ff}=\oplus_{\alpha\in \mathfrak{F}}\mathbb{C}f_{\alpha}$$R$$V^{\mathfrak{F}}\otimes V^{\mathfrak{F}}$, and define$R^{\mathfrak{F}}(\lambda)$, then we call V$\otimes V a finite-

to be the matrix representation

with respect to the basis$\{f_{\alpha}\otimes f_{\beta}\}$.

automatically becomes a solution of the Yang-Baxter

equation (1.1).
Let us construct finite-dimensional representations of the trigonometric $R(\lambda)(4.18)$

in Proposition 4.3 with $A=B=0$ . (We can also obtain finite-dimensional represen-tations of the rational $R(\lambda)(4.19)$ with $A=B=0.$ )Proposition 6.1. For $n\geq 1$ , we set $S=\{0,1, \ldots, n-1\},$$f_{\alpha}(x)=e^{2\pi\sqrt{-1}\alpha x}(\alpha\in ff)$

,

34
and

$V^{\mathfrak{F}}=\oplus_{\alpha\in S}\mathbb{C}f_{\alpha}$. Then$V^{S}\otimes V^{S}$

is a finite-dimensional representation.

Proof. This proposition follows immediately from the action of$u=e^{2\pi\sqrt{-1}\lambda},$ $t=e^{2\pi\sqrt{-1}\kappa}$, and$R(\lambda)$on$V^{\delta}\otimes V^{\theta}$$R^{(n)}( \lambda)=\frac{1}{2\sqrt{-1}}(u^{\frac{1}{2}}-u^{-\frac{1}{2}})(t^{\frac{1}{2}}-t^{-\frac{1}{2}})R(\lambda)$. Set, then$R^{\langle n)}(\lambda)(f_{\alpha}\otimes f_{\beta})$口These finite-dimensional$R$

matrices are new solutions of the Yang-Baxter equation.

But these are not triangular. Now we discuss how to obtain the trigonometric,
triangular$R$matrix from this matrix

Definition 6.2. Let$End(V^{\otimes N})$

(6.2)

$V=\oplus \mathbb{C}v_{\alpha}$$R^{(n)}(\lambda)$.

be a finite-dimensional vector space. For $T\in$, we denote$T(v_{\alpha_{1}} \otimes\cdots\otimes v_{\alpha_{N}})=\sum_{\beta_{1}\ldots\beta_{N}}T_{\beta_{1}\beta_{N}}^{\alpha_{1}.\cdot..\cdot.\alpha_{N}}v_{\beta_{1}}\otimes\cdots\otimes v_{\beta_{N}}$.

35
Then we define(6.3)$T_{\langle P)}\in End(V^{\otimes N})$by

$T_{\langle P)}(v_{\alpha_{1}} \otimes\cdots\otimes v_{\alpha_{N}})=\sum_{w\in \mathfrak{S}_{N}}T_{\alpha_{w^{1}(1)}\ldots\alpha_{w(N)}}^{\alpha\ldots\alpha_{N}}v_{\alpha_{w(1)}}\otimes\cdots\otimes v_{\alpha_{w(N)}}$

which is called the permutation part of$T$

Proposition 6.2. The permutation part$\{f_{\alpha}\}$with respect to the basis$R_{\langle P)}^{(n)}(\lambda)$of$R^{\langle n)}(\lambda)$,$\{v_{\alpha}\}$.

with respect to the basis

satisfies the Yang-Baxter equation (1.1).

Proof. From (1.1),(6.4)$(R_{12}^{\langle n)}(\lambda_{1})R_{13}^{\langle n)}(\lambda_{1}+\lambda_{2})R_{23}^{(n)}(\lambda_{2}))_{(P)}=(R_{23}^{\langle n)}(\lambda_{2})R_{13}^{\langle n)}(\lambda_{1}+\lambda_{2})R_{12}^{(n)}(\lambda_{1}))_{\langle P)}$.

By using (6.1), we easily see that
(6.5)$(R_{12}^{(n)}(\lambda_{1})R_{13}^{(n)}(\lambda_{1}+\lambda_{2})R_{23}^{(n)}(\lambda_{2}))_{(P)}=R_{\langle P)12}^{(n)}(\lambda_{1})R_{(P)13}^{(n)}(\lambda_{1}+\lambda_{2})R_{\langle P)23}^{(n)}(\lambda_{2})$,$(R_{23}^{\langle n)}(\lambda_{2})R_{13}^{\langle n)}(\lambda_{1}+\lambda_{2})R_{12}^{(n)}(\lambda_{1}))_{\langle P)}=R_{\langle P)23}^{(n)}(\lambda_{2})R\{P(\lambda_{1}+\lambda_{2})R_{\langle P)12}^{(n)}(\lambda_{1})$

.

(6.6)

This completes the proof of Proposition 6.2.
The permutation part

$R_{(P)}^{(n)}(\lambda)$

acts on口$V^{\mathfrak{F}}\otimes V^{\mathfrak{F}}$in the following way:

$R\{P)(\lambda)(f_{\alpha}\otimes f_{\beta})$(6.7)$=|u^{-\frac{1}{2}}(t_{\frac{1}{2}}^{\frac{1}{2}}-t^{-\frac{1}{2}})f_{\beta} \otimes f_{\alpha}-t^{\frac{1}{2}}(u_{\frac{1}{2}}^{\frac{1}{2}}-u_{-\frac{1}{2}}^{-\frac{1}{2}})f_{\alpha}\otimes fu(t^{\frac{1}{2}}-t^{-\frac{1}{2}})f_{\beta}^{-\frac{1}{2}}\otimes f_{\alpha}t(u-u)f_{\alpha}\otimes f_{\beta}^{\beta}(u_{\frac{1}{2}}^{-\frac{1}{2}}t-u^{\frac{1}{2}}t)f_{\alpha}\bigotimes_{-}f_{\alpha_{-\frac{1}{2}}}$$(\alpha=\beta)(\alpha>\beta)(\alpha<\beta)$.

36
Therefore$R_{\langle P)}^{(n)}(\lambda)$

is a trigonometric, triangular$R$matrix, that is relavant to the one

given in [1].
Remark 6.1. The idea of finite-dimensional representations of the$R$operator origi-

nates from the paper of Gaudin [8]. We reformulated his idea in an algebraic way.

7. DISCUSSIONSWe propose some issues to be considered hereafter:
The first one is to calculate rigorously the free energy of our vertex model. This
problem may be relavant to the c-functions of the Macdonald’s symmetric functions(cf. [7] [19]). Our original motivations of this study is to construct a two-parameter

deformation of$U(gl_{\infty})$, so it is important to consider the algebra of the$L$opera-

tors associated to our$R$operator. The third problem concerns finite-dimensional

representations of the$R$operator; namely, can we construct finite-dimensional rep-resentations in the elliptic case? Furthermore we think of it interesting to generalize
the

$R$

operator to higher genus case.

ACKNOWLEDGEMENTS
The authors thank Professor M. Jimbo for drawing their attention to the paperby M. Gaudin after they discovered their$R$matrix, Professor J. H. H. Perk for

informing them of several papers related to infinite-dimensional$R$

matrix, Professor

M. Ruiz-Altaba and Professor A. V. Zabrodin for sending their interesting papers to

the authors, and Professor I. V. Cherednik for useful discussions.

37
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1. V. V. Bazhanov, R. M. Kashaev, Cyclic L-operator related with a 3-state R-matrix, preprint

RIMS-702 (1990).
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3. I. V. Cherednik, Soviet Math. Dokl. 33 No.2 (1986).
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6. J. D. Fay, Theta functions on Riemann surfaces, Lecture notes in Mathematics, Springer 1973.
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Baxter models and quantum symmetric spaces, Chicago

preprint EFI 92-11.

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9. C. $G6mez$ , G. Sierra, A new solution to the star-triangle equation based on $U_{q}(sl(2))$ at rootsof unit, to appear in Nucl. Phys. B.

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$A$: Math. Gen. 23 (1990) 1147-1154.

11. C. Jordan, Fonctions elliptiques, Springer 1981.

12. P. P. Kulish, N. Yu. Reshetikhin, E. K. Sklyanin, Lett. Math. Phys. 5 (1981) 393-403.

13. M. Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum
homogeneous spaces, preprint (1992).

14. M. P. Richey, C. A. Tracy, J. Stat. Phys. 423/4 (1986) 311-348.

15. Y. Shibukawa, K. Ueno, in preparation.
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$c^{17}$. K. Ueno, T. Takebayashi, Zonal spherical functions on quantum symmetric spaces and Macdonald’s symmetric polynomials, Lecture Notes in Mathematics 1510 Springer (1992) 142-147.

18. E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge.

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19. A. V. Zabrodin, to appear in Mod. Phys. Lett. A.

E-mail address, Y. Shibukawa: shibu@math.waseda.ac.jp
K. Ueno: ueno@math.waseda.ac.jp