aa r X i v : . [ m a t h - ph ] A p r MATRIX PRODUCT SOLUTIONS TO THE G REFLECTION EQUATION
ATSUO KUNIBA
Abstract
We study the G reflection equation for the three particles in 1 + 1 dimension that undergo a special scat-tering/reflections described by the Pappus theorem. It is a sixth order equation and serves as a natural G analogue of the Yang-Baxter and the reflection equations corresponding to the cubic and the quartic Coxeterrelations of type A and BC , respectively. We construct matrix product solutions to the G reflection equationby exploiting a connection to the representation theory of the quantized coordinate ring A q ( G ).1. Introduction
The Yang-Baxter and the reflection equations R ( α ) R ( α + α ) R ( α ) = R ( α ) R ( α + α ) R ( α ) ,R ( α ) K ( α + α ) R ( α + 2 α ) K ( α ) = K ( α ) R ( α + 2 α ) K ( α + α ) R ( α ) (1)are the fundamental structure in quantum integrable systems in the bulk [1] and at the boundary [4, 22, 12].They are the Yang-Baxterizations (spectral parameter dependent versions) of the cubic and the quartic Coxeterrelations for the simple reflections s , s of the root systems of A and B /C : s s s = s s s , ∆ + = { α , α + α , α } ,s s s s = s s s s , ∆ + = { α , α + α , α + 2 α , α } . Here α , α are the simple roots and ∆ + denotes the set of positive roots which formally correspond to thespectral parameters. They are so ordered that the k th one from the left is s i · · · s i k − ( α i k ) with i k = 1 ( k : odd)and i k = 2 ( k : even). For simplicity we assume R ∈ End( V ⊗ V ) and K ∈ End V for some vector space V .It is natural and by now classic to extend the ‘factorization’ condition like (1) to more general root systems[4]. In this paper we study the G case of the form R ( α ) G ( α + α ) R (2 α + 3 α ) G ( α + 2 α ) R ( α + 3 α ) G ( α )= G ( α ) R ( α + 3 α ) G ( α + 2 α ) R (2 α + 3 α ) G ( α + α ) R ( α ) , (2)where R is a solution to the Yang-Baxter equation by itself and G ∈ End( V ⊗ V ⊗ V ) is the characteristicoperator in the G theory. It is a Yang-Baxterization of the G Coxeter relation s s s s s s = s s s s s s with the spectral parameters corresponding to the positive roots. See (12)–(13).Although the equation (2) was not written down explicitly in [4], it was explained to the author by Cherednik[6] that the G factorization condition is depicted by a three particle scattering diagram corresponding to (2)and it is related to the geometry of the Desargues-Pappus theorem . The equation (2) for generic symbols R and G without assuming a tensor structure on their representation space (i.e. without indices) has appeared asa defining relation of the root algebra of type G [5, Sec.2]. In this paper we call (2) the G reflection equation for simplicity.The purpose of this paper is to construct families of solutions to the G reflection equation with V =( C ) ⊗ n for any positive integer n . Our approach is based on the 3 dimensional (3D) integrability developed in[2, 17, 13, 14] for the Yang-Baxter equation and in [16] for the reflection equation. The most essential idea ofit is to embark on a quantization or a
3D version of the G reflection equation. We introduce the quantized G reflection equation ( L J L J L J ) ◦ F = F ◦ ( J L J L J L ) , (3)which is a G reflection equation (without spectral parameters) up to conjugation by a certain operator F actingon an auxiliary q -boson Fock space. Our finding (Theorem 4.1) is that with a suitable choice of the quantizedscattering amplitude L and J , (3) coincides exactly with the intertwining relation [15, eq.(28)] of the A q ( G ) This is partly described in [4, p982] and will be detailed in Section 2.2. modules labeled by the longest element of the Weyl group [23]. The F corresponds to the intertwiner. Here A q ( g ), for a finite dimensional classical simple Lie algebra g in general, denotes a Hopf subalgebra of the dual U q ( g ) ∗ called quantized coordinate ring . It has been studied from a variety of aspects. See [7, 19, 23, 18, 11, 20,8, 15, 21, 24] for example.In short, we obtain a solution to the quantized G reflection equation (3). It offers a bonus; the equa-tion/solution can be concatenated along the q -boson Fock space for arbitrary n times. The piled n layers of the1 + 1 dimensional scattering diagrams can be viewed as a 3D lattice system in which adjacent layers may beinterchanged locally according to (3) without changing the total statistical weight, a feature roughly referred toas 3D integrability. Anyway, to the n -concatenation of the quantized G reflection equation, one can insert thespectral parameters and evaluate the intertwiner F away appropriately. It brings us back to the original G re-flection equation, thereby producing a solution to it for each n . Actually there are two such recipes called tracereduction and boundary vector reduction. They lead to the solutions ( R tr ( z ) , G tr ( z )) and ( R bv( z ) , G bv ( z )) ,respectively. By the construction they possess the matrix product structure containing n -product of L ’s or J ’s R tr ( z ) = ̺ tr ( z )Tr( z h L · · · L ) , G tr ( z ) = κ tr ( z )Tr( z h J · · · J ) ,R bv ( z ) = ̺ bv ( z ) h ξ | z h L · · · L | ξ i , G bv ( z ) = κ bv ( z ) h ξ | z h J · · · J | ξ i , (4)where the trace and the sandwich h ξ | ( · · · ) | ξ i are taken over a q -boson Fock space. The detail will be explainedin later sections. The solutions are trigonometric in the spectral parameter . In fact R tr ( z ) and R bv ( z ) turnout to be the quantum R matrices [7, 9] for the antisymmetric tensor representations of U p ( A (1) n − ) and the spinrepresentation of U p ( D (2) n +1 ) with p = − q − . This part of the results is contained in the earlier works [2, 17].This paper may be viewed as a continuation of [2, 17] and [16] where analogous results were obtained for theYang-Baxter and the reflection equations, respectively. To explore applications of the G reflection equationis a future problem. For instance to architect commuting transfer matrices based on the G reflection is aninteresting issue.The paper is organized as follows. In Section 2 we explain the interpretation of the G reflection equation interms of a special three particle scattering following [4, 6]. The characteristic feature is the operator G whichencodes the simultaneous reflection of one of the particles at the boundary and scattering of the other two. Theworld-lines of these particles form a configuration matching the classical Pappus theorem.In Section 3 we formulate the quantized G reflection equation by promoting R and G in (2) to the q -bosonvalued L and J . The L matrix (19) appeared first in [2]. The q -boson valued amplitude J (24)–(31) has beendesigned deliberately to validate Theorem 4.1. It does not split into the product of operators representing thesingle particle reflection and the two particle scattering. See (34).In Section 4 after recalling basic facts on the representation theory of A q ( G ) [23], we state our key observationin Theorem 4.1. It identifies the quantized G reflection equation with the intertwining relation between certain A q ( G ) modules.In Section 5 we review the reduction of the tetrahedron equation (cf. [25]) to the Yang-Baxter equationfollowing [2, 17, 16]. This construction has been illustrated in many literatures recently, e.g. [13, 14], so we keepthe description brief. A slightly more detailed exposition is available in [16, App.B].In Section 6 we explain that the analogous reduction works perfectly also for the quantized G reflectionequation. They lead to two families of solutions ( R tr ( z ) , G tr ( z )) and ( R bv ( z ) , G bv ( z )) to the G reflectionequation (2), where the latter is yet based on the conjectural relation (77). The role of the intertwiner F iscurious. Although it is complicated and no closed formula is known, it does not give rise to a difficulty since thereduction procedure just eliminates it. Nevertheless F essentially controls the construction behind the scene inthat it specifies precisely how the L and J are to be combined, how the spectral parameters should be arrangedand what kind of boundary vectors are acceptable. These are essential legacy of F .Section 7 is a summary. Appendix A describes the precise correspondence between the quantized G reflectionequation and the intertwining relation (44) of the A q ( G ) modules. Appendix B contains explicit forms of( R tr ( z ) , G tr ( z )) and ( R bv ( z ) , G bv ( z )) for small n . This latter solution assumes the relation (77) yet to be proved. ̺ tr ( z ) , ̺ bv ( z ) , κ tr ( z ) , κ bv ( z ) are scalars given in (68) and (87). “Trigonometric” means rational in z in (4) which corresponds to the exponential of the spectral parameters in (2). REFLECTION EQUATION 3
Throughout the paper we assume that q is generic and use the following notation:( z ; q ) m = m Y k =1 (1 − zq k − ) , ( q ) m = ( q ; q ) m ,θ (true) = 1 , θ (false) = 0 , e j = (0 , . . . , , j , , . . . , ∈ Z n (1 ≤ j ≤ n ) . G reflection equation for three particle scattering The G reflection equation. Let V be a vector space and consider the operators R ( z ) ∈ End( V ⊗ V ) , G ( z ) ∈ End( V ⊗ V ⊗ V ) (5)depending on the spectral parameter z . We assume that R ( z ) satisfies the Yang-Baxter equation: R ( x ) R ( xy ) R ( y ) = R ( y ) R ( xy ) R ( x ) ∈ End( V ⊗ V ⊗ V ) . (6)By the G reflection equation we mean the following in End( V ⊗ V ⊗ V ): R ( x ) G ( xy ) R ( x y ) G ( xy ) R ( xy ) G ( y )= G ( y ) R ( xy ) G ( xy ) R ( x y ) G ( xy ) R ( x ) . (7)To explain the notation, write temporarily as R ( z ) = P r (1) l ⊗ r (2) l and G ( z ) = P g (1) l ⊗ g (2) l ⊗ g (3) l with somesums over l . Then R ( z ) = X r (1) l ⊗ r (2) l ⊗ , R ( z ) = X r (1) l ⊗ ⊗ r (2) l , R ( z ) = X ⊗ r (1) l ⊗ r (2) l ,R ( z ) = X r (2) l ⊗ r (1) l ⊗ , R ( z ) = X r (2) l ⊗ ⊗ r (1) l , R ( z ) = X ⊗ r (2) l ⊗ r (1) l ,G ijk ( z ) = X g ( i ) l ⊗ g ( j ) l ⊗ g ( k ) l . (8)2.2. Scattering diagram; Pappus configuration.
Let us describe the special three particle scattering relatedto the G reflection equation. This is due to [4, 6]. Consider the three particles 1,2,3 coming from A ,A ,A and being reflected by the boundary at O , O , O , respectively. See Figure 1. The bottom horizontal line isthe boundary which may also be viewed as the time axis. The vertical direction corresponds to the 1D space.Each line carries V which specifies an internal degrees of the freedom of a particle. So a three particle state ata time is described by an element in V ⊗ V ⊗ V .A ✲ A ✲ A ✲ P P P O O O Q Q Q Figure 1.
Scattering diagram for the RHS of (7).One can arrange the three particle world-lines so that the two particle scattering P , P , P happen exactlyat the same instant as the boundary reflection O , O , O of the other particle, respectively. This is nontrivial.For instance, suppose there were only particles 2 and 3. They already determine the reflecting points O , O and the intersection P (and Q ) and its projection O onto the boundary. Let P , P be the points on theworld-lines of particle 3 and 2 whose projection are O and O , respectively. In order to be able to draw the Although these expansions do not specify r ( a ) l , g ( a ) l uniquely, it suffices to make (8) unambiguous. ATSUO KUNIBA world-line for the last particle 1, the three points P , P and O must be collinear. This is guaranteed by aspecial case of the Pappus theorem from the 4th century.One can state it more symmetrically just by starting from P , P and their projection O , O onto theboundary. Let P ′ , P ′ be the mirror image of P , P with respect to the boundary. Then the three intersectionsP O ∩ O P , P P ′ ∩ P ′ P and O P ′ ∩ P ′ O are collinear; in fact they are P , O and the mirror image of P .Let us call the so arranged scattering diagram a Pappus configuration . The reflection at O i with the simul-taneous two particle scattering at P i will be referred to as a special three particle event ( i = 1 , , O and the (dual) reflection angles ∠ P O O and ∠ P O O . Set u = ∠ P O O , w = ∠ P O O , v = ∠ P O O ,θ = ∠ A Q A , θ = ∠ A P A , θ = ∠ A Q O ,θ = ∠ A P O , θ = ∠ A Q O , θ = ∠ O P O . (9)Then it is elementary to seetan w = tan u + tan v, (10) θ = u − v, θ = w − v, θ = u + w, θ = u + v, θ = v + w, θ = w − u. (11)We formally consider the infinitesimal angles hence replace (10) by w = u + v . In such a treatment, a Pappusconfiguration is labeled only by the two angles u and v . By a further substitution u = α + α and v = α , (11)becomes θ = α , θ = α + α , θ = 2 α + 3 α , θ = α + 2 α , θ = α + 3 α , θ = α . (12)Regard the symbols α , α formally as the simple roots of G . They are transformed by the simple reflections s , s of the Weyl group as s ( α ) = − α , s ( α ) = α + α , s ( α ) = α + 3 α , s ( α ) = − α . Thus we find θ k = s i · · · s i k − ( α i k ) , ( i , i , i , i , i , i ) = (1 , , , , , , (13)and { θ , . . . , θ } yields the set of the positive roots of G .The RHS of the G reflection equation (7) is obtained by attaching R (e θ k ) to the two particle scattering atQ i and G (e θ k ) to the special three particle event at P i O i if it is the k th event starting from the left in Figure1. Setting e u = x and e v = y , the assignment reads R ( x ) : two particle scattering at Q ,G ( xy ) : special three particle event at P O ,R ( x y ) : two particle scattering at Q ,G ( xy ) : special three particle event at P O ,R ( xy ) : two particle scattering at Q ,G ( y ) : special three particle event at P O . The indices for each operator correspond to the ordering of the relevant particles before the process. For instancejust before the special three particle event at P O , the incoming particles are 3,1,2 from the top to the bottom,which is encoded in G ( xy ). The LHS of the G reflection equation (7) represents the Pappus configurationin which the time ordering of the processes are reversed. See Figure 2.3. Quantized G reflection equation q -bosons. Let F q = L m ≥ C | m i and F ∗ q = L m ≥ C h m | be the Fock space and its dual equipped with theinner product h m | m ′ i = ( q ) m δ m,m ′ . We define the q -boson operators a + , a − , k on them by a + | m i = | m + 1 i , a − | m i = (1 − q m ) | m − i , k | m i = q m + | m i , h m | a − = h m + 1 | , h m | a + = h m − | (1 − q m ) , h m | k = h m | q m + . (14) REFLECTION EQUATION 5 ✲ ✲ ✲ Figure 2.
Scattering diagram for the LHS of (7).They satisfy ( h m | X ) | m ′ i = h m | ( X | m ′ i ). Let F q , F ∗ q and A + , A − , K denote the same objects with q replacedby q . Namely, A + | m i = | m + 1 i , A − | m i = (1 − q m ) | m − i , K | m i = q m + | m i , h m | A − = h m + 1 | , h m | A + = h m − | (1 − q m ) , h m | K = h m | q m + . The inner product in F q is given by h m | m ′ i = ( q ) m δ m,m ′ differing from the F q case. However we write thebase vectors as h m | , | m i either for F ∗ q , F q or F ∗ q , F q since their distinction will always be evident from thecontext. Note the q -boson commutation relations k a ± = q ± a ± k , a ± a ∓ = 1 − q ∓ k , (15) K A ± = q ± A ± K , A ± A ∓ = 1 − q ∓ K . (16)We will also use the number operator h defined by h | m i = m | m i , h m | h = h m | m (17)either for F q or F q . One may regard k = q h + and K = q h + . The extra 1 / q k is the zero point energy , which simplifies many forthcoming formulas.3.2. q -boson valued L matrix. Set V = C v ⊕ C v ≃ C . This should not be confused with V in (5). Infact they will be related as V = V ⊗ n later. (See around (53).) We introduce the q -boson valued L matrix by L ( v α ⊗ v β ⊗ | m i ) = X γ,δ ∈{ , } v γ ⊗ v δ ⊗ L γ,δα,β | m i , (18) L = (cid:16) L γ,δα,β (cid:17) = K A − A + − K
00 0 0 1 ∈ End( V ⊗ V ⊗ F q ) . (19)We attach a diagram to each component L γ,δα,β ∈ End( F q ) as follows : ✲✻ ✲✻ ✲✻ ✲✻ ✲✻ ✲✻ ✲✻ α β γδ L γ,δα,β K − K A + A − The vertices here should be distinguished from those in Figure 1 and 2 since V and V are different. ATSUO KUNIBA
The other configurations are to be understood as zero. So L may be regarded as defining a q -boson valued sixvertex model in which the latter relation of (16) plays the role of “free-fermion” condition. See eq. (10 . . | d =0 in [1]. Explicitly we have L ( v ⊗ v ⊗ | m i ) = v ⊗ v ⊗ | m i , L ( v ⊗ v ⊗ | m i ) = v ⊗ v ⊗ | m i ,L ( v ⊗ v ⊗ | m i ) = v ⊗ v ⊗ K | m i + v ⊗ v ⊗ A + | m i = q m + v ⊗ v ⊗ | m i + v ⊗ v ⊗ | m + 1 i ,L ( v ⊗ v ⊗ | m i ) = v ⊗ v ⊗ A − | m i − v ⊗ v ⊗ K | m i = (1 − q m ) v ⊗ v ⊗ | m − i − q m + v ⊗ v ⊗ | m i . Note the obvious properties L γ,δα,β = 0 unless α + β = γ + δ, (20) h L γ,δα,β = L γ,δα,β ( h + β − δ ) , (21)which will be referred to as weight conservation . Up to conventional difference, the L matrix (18) appeared in[2]. See also [17, 16].3.3. q -boson valued J matrix. Besides the L , we need another q -boson valued matrix J which encodes acharacteristic feature of the G scattering. It is defined by J ( v α ⊗ v β ⊗ v γ ⊗ | m i ) = X λ,µ,ν ∈{ , } v λ ⊗ v µ ⊗ v ν ⊗ J λ,µ,να,β,γ | m i , (22) J = (cid:16) J λ,µ,να,β,γ (cid:17) ∈ End( V ⊗ V ⊗ V ⊗ F q ) . (23)Each component J λ,µ,να,β,γ ∈ End( F q ) is depicted by a 90 ◦ -degrees rotated special three particle event J λ,µ,να,β,γ = µ λ ν ✣❪ ❪ α β γ (24)We choose the operator J λ,µ,να,β,γ ∈ End( F q ) concretely as follows: a a ✣❪ ❪ a a a + a a ✣❪ ❪ a a k a a ✣❪ ❪ a a − k a a ✣❪ ❪ a a a − ( a = 0 ,
1) (25)1 0 0 ✣❪ ❪ u k a + ✣❪ ❪ k ✣❪ ❪ r − u u s ✣❪ ❪ u k a − (26)0 1 0 ✣❪ ❪ − u a + k ✣❪ ❪ r − u u s ✣❪ ❪ k ✣❪ ❪ − u a − k (27) Note however again that the lines here carry V whereas those in Figure 1 and 2 do V . The boundary line is omitted here. REFLECTION EQUATION 7 ✣❪ ❪ a + ) ✣❪ ❪ u k a + ✣❪ ❪ − u a + k ✣❪ ❪ r − u u s (28)1 0 0 ✣❪ ❪ r − u u s ✣❪ ❪ u k a − ✣❪ ❪ − u a − k ✣❪ ❪ a − ) (29)Here u , u , u , u are parameters satisfying u u + u u = r := q + q − . (30)The operator s ∈ End( F q ) is defined by s = a − a + − q − k = 1 − r k . (31)All the J λ,µ,να,β,γ ’s not contained in the above list is zero. The weight conservation properties analogous to (20)and (21) hold:. J λ,µ,να,β,γ = 0 unless α + β = λ + µ, (32) h J λ,µ,να,β,γ = J λ,µ,να,β,γ ( h + 1 + β − γ − µ − ν ) . (33)As an illustration we have J ( v ⊗ v ⊗ v ⊗ | m i )= v ⊗ v ⊗ v ⊗ J | m i + v ⊗ v ⊗ v ⊗ J | m i + v ⊗ v ⊗ v ⊗ J | m i + v ⊗ v ⊗ v ⊗ J | m i = − u q m + v ⊗ v ⊗ v ⊗ | m + 1 i + r − u u (1 − rq m +1 ) v ⊗ v ⊗ v ⊗ | m i + r − u u (1 − rq m +1 ) v ⊗ v ⊗ v ⊗ | m i + u q m − (1 − q m ) v ⊗ v ⊗ v ⊗ | m − i . The three particle diagram reduces to a direct product of two particle scattering and one particle boundaryreflection if the dotted line were absent. Although it is not the case, the operator J almost splits into such aproduct as J λ,µ,να,β,γ = d L λ,µα,β K νγ + cθ ( α + γ = µ + ν = 1)id (34)for some constants c, d . Here L denotes (19) | A ± → a ± , K → k and K νγ are the q -boson valued K matrix introducedin [16, eq.(9)].3.4. Quantized G reflection equation. Given L and J in Section 3.2 and 3.3, consider the G reflectionequation (7) | R → L,G → J that holds up to conjugation by an element F ∈ End( F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ):( L J L J L J ) ◦ F = F ◦ ( J L J L J L ) . (35)This is an equality of linear operators on V ⊗ V ⊗ V ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q , where the superscriptsare just temporal labels for explanation. If they are all exhibited (35) reads as L J L J L J F = F J L J L J L . (36)We fix the normalization of F by F ( | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ) = | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i . (37) ATSUO KUNIBA
To explain the notation in (35) and (36), write L (18) and J (22) as L = P L (1) l ⊗ L (2) l ⊗ L (3) l and J = P J (1) l ⊗ J (2) l ⊗ J (3) l ⊗ J (4) l similarly to (8), where P means P l . Then L ij = X L ( i ) l ⊗ L ( j ) l ⊗ ⊗ L (3) l ⊗ ⊗ ⊗ ⊗ ⊗ i, j ) = (1 , , (2 , ,L ij = X ⊗ L ( i − l ⊗ L ( j − l ⊗ ⊗ ⊗ L (3) l ⊗ ⊗ ⊗ i, j ) = (2 , , (3 , ,L ij = X L ( i ′ ) l ⊗ ⊗ L ( j ′ ) l ⊗ ⊗ ⊗ ⊗ ⊗ L (3) l ⊗ i, j ) = (1 , , (3 , , ′ = 1 , ′ = 2) ,J ijk = X J ( i ) l ⊗ J ( j ) l ⊗ J ( k ) l ⊗ ⊗ J (4) l ⊗ ⊗ ⊗ ⊗ { i, j, k } = { , , } ) ,J ijk = X J ( i ) l ⊗ J ( j ) l ⊗ J ( k ) l ⊗ ⊗ ⊗ ⊗ J (4) l ⊗ ⊗ { i, j, k } = { , , } ) ,J ijk = X J ( i ) l ⊗ J ( j ) l ⊗ J ( k ) l ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ J (4) l ( { i, j, k } = { , , } ) . Practically, one can realize these operators from (18) and (22) by putting L γ,δα,β and J λ,µ,να,β,γ at appropriate tensorcomponents with a suitable permutations of the indices α, β, . . . . The equation (35) or equivalently (36) isa q -boson valued G reflection equation without a spectral parameter up to conjugation. We call them the quantized G reflection equation in analogy with the quantized reflection equation proposed in [16] for C . It isdepicted as follows.2 ✲ ✲ ✲
579 468 ◦ F = F ◦ ✲ ✲ ✲ q -bosons are acting on them in thedirection perpendicular to this planar diagram. If one introduces such q -boson arrows going from the back tothe front of the diagram, the operator F in the LHS (resp. RHS) corresponds to a vertex where the sixarrows going toward (resp. coming from) 4,5,6,7,8,9 intersect. In Section 6.1 we will take the concatenation of(35) for n times. It corresponds to a 3D diagram involving the n layers of the Pappus configurations depictedin the above.The component of (36) corresponding to the transition v i ⊗ v j ⊗ v k v a ⊗ v b ⊗ v c in V ⊗ V ⊗ V is given by (cid:16)X L a,bα ,α ⊗ J α ,c,α β ,β ,β ⊗ L β ,β γ ,γ ⊗ J γ ,β ,γ λ ,µ ,µ ⊗ L µ ,µ λ ,λ ⊗ J λ ,λ ,λ k,j,i (cid:17) F = F (cid:16)X L α ,α j,i ⊗ J β ,β ,β k,α ,α ⊗ L γ ,γ β ,β ⊗ J µ ,λ ,µ β ,γ ,γ ⊗ L λ ,λ µ ,µ ⊗ J b,c,aλ ,λ ,λ (cid:17) , (38)with the sums taken over α , α , β , β , β , γ , γ , λ , λ , λ , µ , µ ∈ { , } . The summands correspond tovarious diagrams with the external edges specified as j ✲ k ✲ i ✲ abc ◦ F = F ◦ j ✲ k ✲ i ✲ abc Let us illustrate the case ( a, b, c, i, j, k ) = (1 , , , , , REFLECTION EQUATION 9001 1 11 ✲ ✲ ✲
110 11001 1001 ⊗ k ⊗ K ⊗ k ⊗ K ⊗ a −
001 1111 1 ✲ ✲ ✲
01 00 11 0 01 0 A + ⊗ a − ⊗ K ⊗ k ⊗ K ⊗ k
001 1111 1 ✲ ✲ ✲
10 01 01 0 01 0 K ⊗ ( a + ) ⊗ A − ⊗ k ⊗ K ⊗ k
001 1111 1 ✲ ✲ ✲
10 01 11 1 01 0 u u K ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ k
001 1111 1 ✲ ✲ ✲
10 10 00 0 01 0 u u K ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ k
001 1111 1 ✲ ✲ ✲
10 10 10 1 01 0 K ⊗ k ⊗ A + ⊗ ( a − ) ⊗ K ⊗ k
001 1111 1 ✲ ✲ ✲
10 10 11 0 10 0 K ⊗ k ⊗ K ⊗ a + ⊗ A − ⊗ k
001 1111 1 ✲ ✲ ✲
10 10 11 0 11 1 K ⊗ k ⊗ K ⊗ k ⊗ ⊗ a − The the top left diagram yields the LHS while the other ones lead to the RHS. We have also shown thecorresponding q -boson valued amplitude. As the result the quantized G reflection equation (35) in this case becomes the following equation in End( F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ):(1 ⊗ k ⊗ K ⊗ k ⊗ K ⊗ a − ) F = F ( A + ⊗ a − ⊗ K ⊗ k ⊗ K ⊗ k + K ⊗ ( a + ) ⊗ A − ⊗ k ⊗ K ⊗ k + r K ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ k + K ⊗ k ⊗ A + ⊗ ( a − ) ⊗ K ⊗ k + K ⊗ k ⊗ K ⊗ a + ⊗ A − ⊗ k + K ⊗ k ⊗ K ⊗ k ⊗ ⊗ a − ) , (39)where we have combined the terms with coefficients u u and u u together by means of (30).The quantized G reflection equation (35) is the set of 2 = 64 equations like (39) corresponding to thechoice of a, b, c, i, j, k ∈ { , } in (38). In the next section they will be identified with the intertwining relationof certain A q ( G ) modules. 4. A q ( G ) and its intertwiner Intertwining relation of A q ( G ) modules. The quantized coordinate ring A q ( G ) is a Hopf algebrawhich can be realized by 49 generators ( t i,j ) ≤ i,j ≤ obeying the so-called RT T type quadratic relations and someadditional ones. They are available in [20], which was adopted in [15, Sec.3.3.3] in the form directly relevant tothis paper. Their concrete form is not necessary here. What we need is the two fundamental representations π i : A q ( G ) → End( F q i ) associated with the simple reflections s i ( i = 1 , q = q, q = q . (cid:0) π ( t i,j ) (cid:1) ≤ i,j ≤ = a − k − k a + a − ) r k a − k − a − k s k a + k − r k a + ( a + ) a − k − k a + , (cid:0) π ( t i,j ) (cid:1) ≤ i,j ≤ = A − K − K A + A − K
00 0 0 0 − K A +
00 0 0 0 0 0 1 . (40)Here r and s are defined in (30) and (31). These expressions are obtained from [15, eq.(27)] by setting µ = µ = 1, α = q , α = q .The coproduct ∆ ( k ) : A q ( G ) → A q ( G ) ⊗ k takes the simple form ∆ ( k ) ( t i,j ) = P ≤ l ,...,l k ≤ t i,l ⊗ t l ,l ⊗ · · · ⊗ t l k ,j . For i , . . . , i k ∈ { , } , one can construct a tensor product representation by π i ,...,i k := ( π i ⊗ · · · ⊗ π i k ) ◦ ∆ ( k ) : A q ( G ) → End( F q i ⊗ · · · ⊗ F q ik ) . According to the general theory [23], π i ,...,i k is irreducible if and only if s i · · · s i k is a reduced expression of anelement of the Weyl group of G . Moreover π i ,...,i k and π j ,...,j k are equivalent if s i · · · s i k = s j · · · s j k . We areconcerned with the two reduced expressions of the longest element s s s s s s = s s s s s s , the associatedrepresentations π and π and their isomorphism π ≃ π .Let Φ ∨ be the intertwiner. Namely it is the map F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q → F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q characterized by π Φ ∨ = Φ ∨ π up to normalization. Set Φ = Φ ∨ ◦ P where P is a linear map reversingthe order of the six-fold tensor product as P ( x ⊗ x ⊗ · · · ⊗ x ) = x ⊗ x · · · ⊗ x . Thus there exists the uniqueΦ such that Φ ∈ End( F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ) , (41) π ( g ) Φ = Φ π ′ ( g ) ∀ g ∈ A q ( G ) ( π ′ := P π P ) , (42)Φ( | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ) = | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i ⊗ | i . (43) Note that unlike (14) in this paper, the operator k in [15, eq.(17)] does not contain the zero point energy. REFLECTION EQUATION 11
The condition (43) fixes the normalization. It suffices to impose the equation (42) for the 49 generators g = t i,j .By using the explicit form of the coproduct ∆ (6) , they are expressed as (cid:16)X π ( t i,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,j ) (cid:17) Φ= Φ (cid:16)X π ( t l ,j ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t l ,l ) ⊗ π ( t i,l ) (cid:17) (1 ≤ i, j ≤ , (44)where the sums are taken over 1 ≤ l , . . . , l ≤
7. In this way the intertwining relation (42) boils down to the 49equations (44). Although the lists of π ( t i,j ) , π ( t i,j ) in (40) are pretty sparse, some equations become lengthyincluding typically 16 terms on one side or both. We do not display them all here but present a few examples. g = t , :(1 ⊗ a − ⊗ ⊗ a − ⊗ ⊗ a − − ⊗ a − ⊗ ⊗ k ⊗ A − ⊗ k − ⊗ k ⊗ A − ⊗ a + ⊗ A − ⊗ k − ⊗ k ⊗ A − ⊗ k ⊗ ⊗ a − + 1 ⊗ k ⊗ K ⊗ ( a − ) ⊗ K ⊗ k )Φ= Φ(1 ⊗ a − ⊗ ⊗ a − ⊗ ⊗ a − − ⊗ a − ⊗ ⊗ k ⊗ A − ⊗ k − ⊗ k ⊗ A − ⊗ a + ⊗ A − ⊗ k − ⊗ k ⊗ A − ⊗ k ⊗ ⊗ a − + 1 ⊗ k ⊗ K ⊗ ( a − ) ⊗ K ⊗ k ) ,g = t , :(1 ⊗ a − ⊗ ⊗ k ⊗ K ⊗ k + 1 ⊗ k ⊗ A − ⊗ a + ⊗ K ⊗ k + 1 ⊗ k ⊗ K ⊗ ( a − ) ⊗ A + ⊗ k + r ⊗ k ⊗ K ⊗ k a − ⊗ ⊗ k a + + 1 ⊗ k ⊗ K ⊗ k ⊗ A − ⊗ ( a + ) )Φ= Φ( A − ⊗ ( a + ) ⊗ A − ⊗ k ⊗ K ⊗ k + r A − ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ k + A − ⊗ k ⊗ A + ⊗ ( a − ) ⊗ K ⊗ k + A − ⊗ k ⊗ K ⊗ a + ⊗ A − ⊗ k + A − ⊗ k ⊗ K ⊗ k ⊗ ⊗ a − − K ⊗ a − ⊗ K ⊗ k ⊗ K ⊗ k ) ,g = t , :(1 ⊗ k ⊗ K ⊗ k ⊗ K ⊗ a − )Φ= Φ( A + ⊗ a − ⊗ K ⊗ k ⊗ K ⊗ k + K ⊗ ( a + ) ⊗ A − ⊗ k ⊗ K ⊗ k + r K ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ k + K ⊗ k ⊗ A + ⊗ ( a − ) ⊗ K ⊗ k + K ⊗ k ⊗ K ⊗ a + ⊗ A − ⊗ k + K ⊗ k ⊗ K ⊗ k ⊗ ⊗ a − ) , (45) g = t , :( A − ⊗ a + ⊗ K ⊗ k ⊗ K ⊗ a − + K ⊗ ( a − ) ⊗ A + ⊗ k ⊗ K ⊗ a − + r K ⊗ k a − ⊗ ⊗ k a + ⊗ K ⊗ a − + K ⊗ k ⊗ A − ⊗ ( a + ) ⊗ K ⊗ a − + K ⊗ k ⊗ K ⊗ a − ⊗ A + ⊗ a − − K ⊗ k ⊗ K ⊗ k ⊗ ⊗ k )Φ= Φ( A + ⊗ a − ⊗ K ⊗ k ⊗ K ⊗ a + + K ⊗ ( a + ) ⊗ A − ⊗ k ⊗ K ⊗ a + + r K ⊗ k a + ⊗ ⊗ k a − ⊗ K ⊗ a + + K ⊗ k ⊗ A + ⊗ ( a − ) ⊗ K ⊗ a + + K ⊗ k ⊗ K ⊗ a + ⊗ A − ⊗ a + − K ⊗ k ⊗ K ⊗ k ⊗ ⊗ k ) ,g = t , , t , :[Φ , ⊗ k ⊗ K ⊗ k ⊗ K ⊗ k ] = 0 . Solution to the quantized G reflection equation. Notice that (45) coincides with (39) under theidentification F = Φ. In fact under this correspondence one can directly check that the 49 intertwining relations(44) and the 64 quantized G reflection equations (38) are equivalent . We list the correspondence of the indices( i, j ) in (44) and ( a, b, c, i, j, k ) in (38) in Appendix A. Since (43) and (37) impose the same normalization on Φand F , we conclude F = Φ. Let us summarize this result in Theorem 4.1.
Under the normalization (37), the quantized G reflection equation (35) with L, J given in Sec-tion 3.2 and 3.3 has the unique solution F = Φ in terms of the intertwiner Φ of the A q ( G ) module characterizedby (42) and (43). Henceforth we shall identify F and Φ and write F to also mean the intertwiner Φ. Let us quote some basicproperties of F from [15, Sec.4.4]. Set F ( | i i ⊗ | j i ⊗ | k i ⊗ | l i ⊗ | m i ⊗ | n i ) = X a,b,c,d,e,f ∈ Z ≥ F abcdefijklmn | a i ⊗ | b i ⊗ | c i ⊗ | d i ⊗ | e i ⊗ | f i . Then the following properties are valid: F abcdefijklmn ∈ Z [ q ] , F abcdefijklmn = 0 unless (cid:18) a + b +2 c + d + eb +3 c +2 d +3 e + f (cid:19) = (cid:18) i + j +2 k + l + mj +3 k +2 l +3 m + n (cid:19) , (46) F − = F , F abcdefijklmn = ( q ) i ( q ) j ( q ) k ( q ) l ( q ) m ( q ) n ( q ) a ( q ) b ( q ) c ( q ) d ( q ) e ( q ) f F ijklmnabcdef . (47)Due to the latter property of (46), F is an infinite direct sum of finite dimensional matrices. In terms of h i acting as h (17) on the i th component from the left, it may be rephrased as the commutativity[ F , x h ( xy ) h ( x y ) h ( xy ) h ( xy ) h y h ] = 0 , (48)where x and y are free parameters. We let F also act on h ω | ∈ F ∗ q ⊗ F ∗ q ⊗ F ∗ q ⊗ F ∗ q ⊗ F ∗ q ⊗ F ∗ q by ( h ω | F ) | ω ′ i = h ω | ( F | ω ′ i ) for any | ω ′ i ∈ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q .It is possible to make a tedious computer program to calculate F abcdefijklmn for any given indices by using (44).However unlike the A q ( A ) and A q ( C ) cases, an explicit general formula is yet to be constructed. At q = 0 F abcdefijklmn is known to become 0 or 1, which can be determined by the ultradiscretization (tropical form) of [3,Th.3.1(c)]. Example 4.2.
The following is the list of all the nonzero F abcdef . F = − q (1 − q )(1 − q )(1 − q − q ) , F = − q (1 − q ) (1 − q )(1 + q ) , F = (1 − q )(1 − q )(1 − q + q ) , F = q , F = q (1 − q )(1 − q − q − q − q − q ) , F = q (1 − q )(1 − q + q ) , F = q (1 + q − q − q − q + q + q + q ) , F = q (1 + q )(1 − q − q ) , F = q (1 + q )(1 − q )(1 − q − q − q + q + q + q ) . Matrix product solutions to Yang-Baxter equation from L Tetrahedron equation.
The q -boson valued L matrix (19) is known to satisfy a version the tetrahedronequation [2] L L L R = R L L L ∈ End( V ⊗ V ⊗ V ⊗ F q ⊗ F q ⊗ F q ) . (49)This is a Yang-Baxter equation up to conjugation by R ∈ End( F q ⊗ F q ⊗ F q ). Such an R is unique up tooverall normalization and is known to satisfy the tetrahedron equation R R R R = R R R R among themselves. See [10] for the approach from the representation theory of the quantized coordinate ring A q ( A ), [2] for a quantum geometry argument and [16, Sec.3.1] for a brief guide to the background.We let R also act on ( F ∗ q ) ⊗ by (cid:0) ( h a | ⊗ h b | ⊗ h c | ) R (cid:1) ( | i i ⊗ | j i ⊗ | k i ) = ( h a | ⊗ h b | ⊗ h c | ) (cid:0) R ( | i i ⊗ | j i ⊗ | k i ) (cid:1) . Inthis paper we will only need the following properties R − = R , [ R , x h + h y h + h ] = 0 , (50)( h χ | ⊗ h χ | ⊗ h χ | ) R = h χ | ⊗ h χ | ⊗ h χ | , R ( | χ i ⊗ | χ i ⊗ | χ i ) = | χ i ⊗ | χ i ⊗ | χ i , (51) h χ | = X m ≥ h m | ( q ) m ∈ F ∗ q , | χ i = X m ≥ | m i ( q ) m ∈ F q , (52)where x, y are free parameters. The relation (51) was proved in [17, Pr.4.1] .5.2. Concatenation of the tetrahedron equation.
Consider n copies of (49) in which the spaces labeledwith 1 , , i , i , i with i = 1 , , . . . , n :( L i i L i i L i i ) R = R ( L i i L i i L i i ) . Sending R to the left by repeatedly applying this relation, we get( L L L ) · · · ( L n n L n n L n n ) R = R ( L L L ) · · · ( L n n L n n L n n ) . (53) χ is just a symbol and not a nonnegative integer. A similar caution applies also to ξ in (76). REFLECTION EQUATION 13
Set V = V ⊗ n ≃ ( C ) ⊗ n in general and k V = k V ⊗ · · · ⊗ k n V when the label is present. The equality (53) holdsin End( V ⊗ V ⊗ V ⊗ F q ⊗ F q ⊗ F q ). It is possible to rearrange it without changing the order of any twooperators sharing common labels as( L · · · L n n )( L · · · L n n )( L · · · L n n ) R = R ( L · · · L n n )( L · · · L n n )( L · · · L n n ) . (54)Write the right relation in (50) as R − x h ( xy ) h y h = x h ( xy ) h y h R − . Multiplying this to (54) from the leftwe get R − (cid:0) x h L · · · L n n (cid:1)(cid:0) ( xy ) h L · · · L n n (cid:1)(cid:0) y h L · · · L n n (cid:1) R = (cid:0) y h L · · · L n n (cid:1)(cid:0) ( xy ) h L · · · L n n (cid:1)(cid:0) x h L · · · L n n (cid:1) . (55)5.3. Reduction to Yang-Baxter equation.
The trace of (55) over F q ⊗ F q ⊗ F q givesTr ( x h L · · · L n n )Tr (( xy ) h L · · · L n n )Tr ( y h L · · · L n n )= Tr ( y h L · · · L n n )Tr (( xy ) h L · · · L n n )Tr ( x h L · · · L n n ) . (56)Alternatively one may sandwich (55) between the bra vector ( h χ | ⊗ h χ | ⊗ h χ | ) and the ket vector | χ i ⊗ | χ i ⊗ | χ i .From R − = R (50) and (51), the result becomes h χ | x h L · · · L n n | χ ih χ | ( xy ) h L · · · L n n | χ ih χ | y h L · · · L n n | χ i = h χ | y h L · · · L n n | χ ih χ | ( xy ) h L · · · L n n | χ ih χ | x h L · · · L n n | χ i . (57)Set R tr1 , ( z ) = ̺ tr ( z )Tr a ( z h a L a · · · L n n a ) ∈ End( V ⊗ V ) , (58) R bv1 , ( z ) = ̺ bv ( z ) h a χ | z h a L a · · · L n n a | a χ i ∈ End( V ⊗ V ) , (59)where a is a dummy label for the auxiliary Fock space a F q . The normalization factors ̺ tr ( z ) and ̺ br ( z ) will bespecified in (68). Now (56) and (57) are both stated as the Yang-Baxter equation R ( x ) R ( xy ) R ( y ) = R ( y ) R ( xy ) R ( x ) (60)with R ( z ) = R tr ( z ) and R bv ( z ). We call the above procedure to get the solutions R tr ( z ) and R bv ( z ) of theYang-Baxter equation from the tetrahedron equation (49) the trace reduction and the boundary vector reduction ,respectively. The vectors (52) are referred to as boundary vectors .The trace reduction is due to [2] and the boundary vector reduction in this paper is a special case of moregeneral ones in [17]. The solutions R tr ( z ) and R bv ( z ) have been identified with the quantum R matrices for theantisymmetric tensor representations of U p ( A (1) n − ) and the spin representation of U p ( D (2) n +1 ) with p = − q − . Aconcise summary of these results can be found in [16, App.B] .5.4. Basic properties of R tr ( z ) and R bv ( z ). We write the base vectors of V = V ⊗ n as | α i = v α ⊗ · · ·⊗ v α n in terms of an array α = ( α , . . . , α n ) ∈ { , } n . We warn that | α i ∈ V should not be confused with the base | m i of a Fock space containing a single nonnegative integer. Set R ( z )( | α i ⊗ | β i ) = X γ , δ ∈{ , } n R ( z ) γ , δα , β | γ i ⊗ | δ i ( R = R tr , R bv ) . Then (58) and (59) imply the matrix product formulas as R tr ( z ) γ , δα , β = ̺ tr ( z )Tr (cid:0) z h L γ ,δ α ,β · · · L γ n ,δ n α n ,β n (cid:1) , (61) R bv ( z ) γ , δα , β = ̺ bv ( z ) h χ | z h L γ ,δ α ,β · · · L γ n ,δ n α n ,β n | χ i , (62) R bv ( z ) corresponds to S , ( z ) in [16]. Since L in [16] is q -boson valued, the formulas in [16, App.B] fit this paper if q thereis replaced by q . where L γ,δα,β is given by (19) and Tr( · · · ) and h χ | ( · · · ) | χ i are taken over F q . They are evaluated by using thecommutation relations (16), the formula (86) | q → q andTr( z h K r ( A + ) s ( A − ) s ′ ) = δ s,s ′ q r ( q ; q ) s ( zq r ; q ) s +1 . (63)For α = ( α , . . . , α n ) ∈ { , } n set | α | = α + · · · + α n , V k = M α ∈{ , } n , | α | = k C | α i . (64)By the definition the direct sum decomposition V = V ⊕ V ⊕ · · · ⊕ V n holds. From (20), (21) and (52) onecan show R ( z ) γ , δα , β = 0 unless α + β = γ + δ ∈ Z n ( R = R tr , R bv ) , (65) R tr ( z ) γ , δα , β = 0 unless | α | = | γ | and | β | = | δ | . (66)The property (66) implies the decomposition R tr ( z ) = M ≤ l,m ≤ n R tr l,m ( z ) , R tr l,m ( z ) ∈ End( V l ⊗ V m ) . (67)The Yang-Baxter equation (60) with R ( z ) = R tr ( z ) is valid for each subspace V k ⊗ V l ⊗ V m of V ⊗ V ⊗ V .The scalar prefactor in (58) for the summand R tr l,m ( z ) in (67) may be taken as ̺ tr l,m ( z ) depending on l and m .We choose it and the one in (59) as ̺ tr l,m ( z ) = q − | l − m | (1 − zq | l − m | ) , ̺ bv ( z ) = ( z ; q ) ∞ ( − zq ; q ) ∞ . (68)They make all the matrix elements of R tr l,m ( z ) and R bv ( z ) rational in z and q . For example we have R tr l,m ( z )( | e [1 ,l ] i ⊗ | e [1 ,m ] i ) = ( − max( l − m, | e [1 ,l ] i ⊗ | e [1 ,m ] i ,R bv ( z )( | i ⊗ | i ) = | i ⊗ | i , (69)where e [1 ,k ] = e + · · · + e k and | i = | , , . . . , i .6. Reduction of quantized G reflection equation Starting from the quantized G reflection equation (36), one can perform two kinds of reductions similar toSection 5 to construct solutions to the G reflection equation (7) in the matrix product form. This is the mainresult of the paper which we are going to present in this section.6.1. Concatenation of quantized G reflection equation. Consider n copies of (36) in which the spaceslabeled with 1 , , i , i , i with i = 1 , , . . . , n :( L i i J i i i L i i J i i i L i i J i i i ) F = F ( J i i i L i i J i i i L i i J i i i L i i ) . (70)Using (70) successively, one can bring F to the left to derive( L J L J L J ) · · · ( L n n J n n n L n n J n n n L n n J n n n ) F = F ( J L J L J L ) · · · ( J n n n L n n J n n n L n n J n n n L n n ) . This can be rearranged without changing the order of operators sharing common labels as( L · · · L n n )( J · · · J n n n )( L · · · L n n ) × ( J · · · J n n n )( L · · · L n n )( J · · · J n n n ) F = F ( J · · · J n n n )( L · · · L n n )( J · · · J n n n ) × ( L · · · L n n )( J · · · J n n n )( L · · · L n n ) . (71)Write (48) as F − x h ( xy ) h ( x y ) h ( xy ) h ( xy ) h y h = y h ( xy ) h ( xy ) h ( x y ) h ( xy ) h x h F − REFLECTION EQUATION 15 and multiply it to (71) from the left. The result reads F − (cid:0) x h L · · · L n n (cid:1)(cid:0) ( xy ) h J · · · J n n n (cid:1)(cid:0) ( x y ) h L · · · L n n (cid:1) × (cid:0) ( xy ) h J · · · J n n n (cid:1)(cid:0) ( xy ) h L · · · L n n (cid:1)(cid:0) y h J · · · J n n n (cid:1) F = (cid:0) y h J · · · J n n n (cid:1)(cid:0) ( xy ) h L · · · L n n (cid:1)(cid:0) ( xy ) h J · · · J n n n (cid:1) × (cid:0) ( x y ) h L · · · L n n (cid:1)(cid:0) ( xy ) h J · · · J n n n (cid:1)(cid:0) x h L · · · L n n (cid:1) . (72)6.2. Trace reduction.
Taking the trace of (72) over F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q ⊗ F q , we obtainTr (cid:0) x h L · · · L n n (cid:1) Tr (cid:0) ( xy ) h J · · · J n n n (cid:1) Tr (cid:0) ( x y ) h L · · · L n n (cid:1) × Tr (cid:0) ( xy ) h J · · · J n n n (cid:1) Tr (cid:0) ( xy ) h L · · · L n n (cid:1) Tr (cid:0) y h J · · · J n n n (cid:1) = Tr (cid:0) y h J · · · J n n n (cid:1) Tr (cid:0) ( xy ) h L · · · L n n (cid:1) Tr (cid:0) ( xy ) h J · · · J n n n (cid:1) × Tr (cid:0) ( x y ) h L · · · L n n (cid:1) Tr (cid:0) ( xy ) h J · · · J n n n (cid:1) Tr (cid:0) x h L · · · L n n (cid:1) . (73)Here Tr ( · · · ) , Tr ( · · · ) , Tr ( · · · ) are identified with R tr ( z ) in (58). The other factors emerging from J have theform G tr123 ( z ) = κ tr ( z )Tr a (cid:0) z h a J a · · · J n n n a (cid:1) ∈ End( V ⊗ V ⊗ V ) , (74)where k V = k V ⊗ · · · ⊗ k n V ≃ ( C ) ⊗ n as before. The trace is taken over a F q and evaluated by means of (15) and(63) | q → q / . The scalar κ tr ( z ) will be specified in (87). Now the relation (73) is rephrased as R tr12 ( x ) G tr132 ( xy ) R tr23 ( x y ) G tr213 ( xy ) R tr31 ( xy ) G tr321 ( y )= G tr231 ( y ) R tr13 ( xy ) G tr123 ( xy ) R tr32 ( x y ) G tr312 ( xy ) R tr21 ( x ) . (75)Thus the pair ( R tr ( z ) , G tr ( z )) yields a solution to the G reflection equation (7).6.3. Boundary vector reduction.
Set h ξ | = X m ≥ h m | ( q ) m ∈ F ∗ q , | ξ i = X m ≥ | m i ( q ) m ∈ F q , (76)which are formally the boundary vectors (52) with q replaced by q . Supported by computer experiments weconjecture ( h χ | ⊗ h ξ | ⊗ h χ | ⊗ h ξ | ⊗ h χ | ⊗ h ξ | ) F = h χ | ⊗ h ξ | ⊗ h χ | ⊗ h ξ | ⊗ h χ | ⊗ h ξ | , F ( | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i ) = | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i , (77)where h χ | and | χ i are defined in (52). Sandwich the relation (72) between the bra vector h χ | ⊗ h ξ | ⊗ h χ | ⊗ h ξ | ⊗h χ | ⊗ h ξ | and the ket vector | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i ⊗ | χ i ⊗ | ξ i . Thanks to (77) and the left relation in (47), theresult becomes h χ | x h L · · · L n n | χ ih ξ | ( xy ) h J · · · J n n n | ξ ih χ | ( x y ) h L · · · L n n | χ i× h ξ | ( xy ) h J · · · J n n n | ξ ih χ | ( xy ) h L · · · L n n | χ ih ξ | y h J · · · J n n n | ξ i = h ξ | y h J · · · J n n n | ξ ih χ | ( xy ) h L · · · L n n | χ ih ξ | ( xy ) h J · · · J n n n | ξ i× h χ | ( x y ) h L · · · L n n | χ ih ξ | ( xy ) h J · · · J n n n | ξ ih χ | x h L · · · L n n | χ i . (78)The factors h χ | ( · · · ) | χ i involving L are identified with R bv ( z ) in (59). The other factors emerging from J havethe form G bv123 ( z ) = κ bv ( z ) h a ξ | z h a J a · · · J n n n a | a ξ i ∈ End( V ⊗ V ⊗ V ) , (79)where the scalar κ bv ( z ) will be specified in (87). In terms of (79) and (62), the relation (78) is stated as R bv12 ( x ) G bv132 ( xy ) R bv23 ( x y ) G bv213 ( xy ) R bv31 ( xy ) G bv321 ( y )= G bv231 ( y ) R bv13 ( xy ) G bv123 ( xy ) R bv32 ( x y ) G bv312 ( xy ) R bv21 ( x ) . (80) The two relations in (77) are actually equivalent due to the right property in (47).
Thus the pair ( R bv ( z ) , G bv ( z )) provides another solution to the G reflection equation (7).6.4. Basic properties of G tr ( z ) and G bv ( z ). The construction (74) and (79) imply the matrix productformula for each element as G ( z )( | α i ⊗ | β i ⊗ | γ i ) = X λ , µ , ν ∈{ , } n G ( z ) λ , µ , να , β , γ | λ i ⊗ | µ i ⊗ | ν i ( G = G tr , G bv ) , (81) G tr ( z ) λ , µ , να , β , γ = κ tr ( z )Tr (cid:0) z h J λ ,µ ,ν α ,β ,γ · · · J λ n ,µ n ,ν n α n ,β n ,γ n (cid:1) , (82) G bv ( z ) λ , µ , να , β , γ = κ bv ( z ) h ξ | z h J λ ,µ ,ν α ,β ,γ · · · J λ n ,µ n ,ν n α n ,β n ,γ n | ξ i (83)in terms of J λ,µ,να,β,γ specified in (24)–(31). From (32) and (33) one can show G tr ( z ) λ , µ , να , β , γ = 0 unless α + β = λ + µ ∈ Z n and n + | β | − | γ | = | µ | + | ν | (84)or equivalently the direct sum decomposition: G tr ( z ) = M l,m,k G tr ( z ) l,m,k , G tr ( z ) l,m,k : V l ⊗ V m ⊗ V k → M k ′ V l + k + k ′ − n ⊗ V m − k − k ′ + n ⊗ V k ′ , (85)where the sums extend over l, m, k, k ′ ∈ [0 , n ] such that the indices l + k + k ′ − n and m − k − k ′ + n also belongto [0 , n ].The trace (82) is evaluated by means of (63) with q replaced by q . The quantity h ξ | ( · · · ) | ξ i in (83) iscalculated from (15) and h ξ | z h ( a ± ) j k m | ξ i = q m ( − q ; q ) j ( − q j + m +1 z ; q ) ∞ ( q m z ; q ) ∞ × ( z j ,q mj , (86)which is easily derived by only using the elementary identity P j ≥ w ; q ) j ( q ; q ) j z j = ( wz ; q ) ∞ ( z ; q ) ∞ . We choose the normal-ization factors in (82) and (83) as κ tr ( z ) = 1 , κ bv ( z ) = ( z ; q ) ∞ ( − qz ; q ) ∞ . (87)Then all the matrix elements (82) and (83) become rational in z and q . For instance we have G tr ( z )( | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i ) = ( q ) m − l + n − zq m − l + n | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i + · · · ( l ≤ m ) , (88) G tr ( z )( | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i ) = ( − q ) l − m + n − zq l − m + n | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i + · · · ( l ≥ m ) , (89) G bv ( z )( | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i ) = q m − l + n ( z ; q ) m − l + n ( − qz ; q ) m − l + n | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i + · · · ( l ≤ m ) , (90) G bv ( z )( | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i ) = ( − q ) l − m + n ( z ; q ) l − m + n ( − qz ; q ) l − m + n | e [1 ,l ] i ⊗ | e [1 ,m ] i ⊗ | i + · · · ( l ≥ m ) , (91)where the symbol e [1 ,k ] was defined after (69) and | i = | e [1 ,n ] i = | , , . . . , i . Remark 6.1.
Using (67) and (85) it is easy to see that the both sides of (75) applied to V s ⊗ V t ⊗ V u generate the space L t ′ V t − s + t ′ ⊗ V t ′ ⊗ V n − u − t − t ′ . There are three R tr ( z )’s on each side of (75). One cancheck that changing their normalization as R tr l,m ( z ) → φ l,m ( z ) R tr l,m ( z ) depending on l, m in (67) keeps (75)valid for any function φ l,m ( z ) of the form ˜ φ l − m ( z ). In fact the both sides acquire the common overall factor˜ φ s + u − n ( xy ) ˜ φ t + u − n ( x y ) ˜ φ t − s ( x ) under the change. Our ̺ tr l,m ( z ) (68) is of this form, hence (75) remains validdespite the “mixing” of weights (or indices) in (85).7. Summary
We have studied the G reflection equation (7) which is a natural G analogue of the Yang-Baxter and thereflection equations corresponding to the A and B /C Coxeter relations, respectively. It describes the threeparticle scattering/reflections whose world-lines form a Pappus configuration. We introduced the quantized G reflection equation (35). It is a q -boson valued G reflection equation that holds up to conjugation. We gavea solution to it in Theorem 4.1 by exploiting a connection to the quantized coordinate ring A q ( G ) [15]. Fromthe concatenation of the solution we have constructed matrix product solutions to the original G reflectionequation ( R tr ( z ) , G tr ( z )) in (75) and ( R bv ( z ) , G bv ( z )) in (80), where the latter assumes the conjecture (77). REFLECTION EQUATION 17
The R and G matrices are linear operators on V ⊗ V and V ⊗ V ⊗ V with V ≃ ( C ) ⊗ n and trigonometric in thespectral parameter. The special three particle event characteristic to the G theory is encoded in G tr ( z ) , G bv ( z )whereas the companion R matrices R tr ( z ) , R bv ( z ) for the two particle scattering are the known ones for theantisymmetric tensor representations of U p ( A (1) n − ) and the spin representations of U p ( D (2) n +1 ) with p = − q − . Appendix A. Correspondence between (38) and (44)
We indicate the equivalence of (38) with ( a, b, c, i, j, k ) ∈ { , } and (44) with ( i, j ) → ( i ′ , j ′ ) ∈ { , . . . , } as ( abcijk ; i ′ j ′ ). In this notation the example treated in (39) and (45) is (111100; 16). There is no need to takelinear combinations etc of the equations and the equivalence literally means the same equation up to an overallconstant. (000000; 77) , (000001; 74) , (000010; 75) , (000011; 72) , (000100; 76) , (000101; 73) , (000110; 74) , (000111; 71) , (001000; 47) , (001001; 44) , (001010; 45) , (001011; 42) , (001100; 46) , (001101; 43) , (001110; 44) , (001111; 41) , (010000; 57) , (010001; 54) , (010010; 55) , (010011; 52) , (010100; 56) , (010101; 53) , (010110; 54) , (010111; 51) , (011000; 27) , (011001; 24) , (011010; 25) , (011011; 22) , (011100; 26) , (011101; 23) , (011110; 24) , (011111; 21) , (100000; 67) , (100001; 64) , (100010; 65) , (100011; 62) , (100100; 66) , (100101; 63) , (100110; 64) , (100111; 61) , (101000; 37) , (101001; 34) , (101010; 35) , (101011; 32) , (101100; 36) , (101101; 33) , (101110; 34) , (101111; 31) , (110000; 47) , (110001; 44) , (110010; 45) , (110011; 42) , (110100; 46) , (110101; 43) , (110110; 44) , (110111; 41) , (111000; 17) , (111001; 14) , (111010; 15) , (111011; 12) , (111100; 16) , (111101; 13) , (111110; 14) , (111111; 11) . Note for instance that (000001; 74) and (000110; 74) imply that not all of the quantized reflection equations (38)are independent.
Appendix B. Examples
Recall the notation | α i ∈ V = V ⊗ n ≃ ( C ) ⊗ n declared in the beginning of Section 5.4. We write | α i ⊗ | β i ∈ V ⊗ V and | α i ⊗ | β i ⊗ | γ i ∈ V ⊗ V ⊗ V with α = ( α , . . . , α n ) ∈ { , } n etc as | α . . . α n , β . . . β n i and | α . . . α n , β . . . β n , γ . . . γ n i , respectively.B.1. R tr ( z ) and G tr ( z ) with n = 1. R tr ( z ) is given by | i, j i 7→ ( − i (1 − j ) | i, j i ( i, j = 0 , G tr ( z ) is nontrivial even for n = 1. Its action on V ⊗ = V ⊗ is given by | , , i 7→ q | , , i − qz , | , , i 7→ − q | , , i − qz , | , , i 7→ q | , , i − q z , | , , i 7→ − u u ( q − z ) | , , i r (1 − z )(1 − q z ) − u u ( q − z ) | , , i r (1 − z )(1 − q z ) , | , , i 7→ − u u ( q − z ) | , , i r (1 − z )(1 − q z ) − u u ( q − z ) | , , i r (1 − z )(1 − q z ) , | , , i 7→ q | , , i − q z , | , , i 7→ q | , , i − qz , | , , i 7→ − q | , , i − qz , where r, u , u , u , u are to obey (30). The two kinds of the denominators 1 − qz and 1 − q z originate in J , , , , = k and J , , , , = k .B.2. R tr ( z ) and G tr ( z ) with n = 2. R tr l,m ( z ) (0 ≤ l, m ≤
2) is the identity except R tr1 , ( z ) = − id, R tr2 , ( z ) = − id and R tr1 , ( z ). The last one R tr1 , ( z ) is given by | ij, ij i 7→ | ij, ij i ( i = 1 − j = 0 , , | , i 7→ − q (1 − z ) | , i − q z + (1 − q ) z | , i − q z , | , i 7→ (1 − q ) | , i − q z − q (1 − z ) | , i − q z , which is a six-vertex. As for G tr ( z ), it is too lengthy to present all the data. So we give just a few examples. | , , i 7→ q | , , i − q z , | , , i 7→ (1 − q ) z | , , i (1 − z )(1 − q z ) − q | , , i − q z , | , , i 7→ q u u ( q − z ) | , , i r (1 − qz )(1 − q z ) − q (1 − q ) u z | , , i (1 − qz )(1 − q z ) + q u u ( q − z ) | , , i r (1 − qz )(1 − q z ) , | , , i 7→ u u u ( q + z − q z − q z + q z + q z ) | , , i ) r (1 − z )(1 − q z )(1 − q z )+ u u u u ( q + z − q z − q z + q z + q z ) | , , i r (1 − z )(1 − q z )(1 − q z ) − q (1 − q ) u u | , , i (1 − q z )(1 − q z ) − q (1 − q ) u u z | , , i (1 − q z )(1 − q z )+ u u u u ( q + z − q z − q z + q z + q z ) | , , i r (1 − z )(1 − q z )(1 − q z )+ u u u ( q + z − q z − q z + q z + q z ) | , , i r (1 − z )(1 − q z )(1 − q z ) . B.3. R bv ( z ) and G bv ( z ) with n = 1. R bv ( z ) reduces to another six-vertex model: | i, i i 7→ | i, i i ( i = 0 , , | , i 7→ q (1 − z ) | , i q z + (1 + q ) z | , i q z , | , i 7→ (1 + q ) | , i q z − q (1 − z ) | , i q z .G bv ( z ) is given by | , , i 7→ (1 + q ) z | , , i qz + q (1 − z ) | , , i qz , | , , i 7→ − q (1 − z ) | , , i qz + (1 + q ) | , , i qz , | , , i 7→ q (1 + q ) u (1 − z ) z | , , i (1 + qz )(1 + q z ) + q (1 − z )(1 − qz ) | , , i (1 + qz )(1 + q z )+ (1 + q )(1 + q ) z | , , i (1 + qz )(1 + q z ) + q (1 + q ) u (1 − z ) z | , , i (1 + qz )(1 + q z ) , | , , i 7→ u ( − q + z + 2 qz + 2 q z + q z − qz )( u | , , i + u | , , i ) r (1 + qz )(1 + q z )+ q (1 + q ) u (1 − z )( | , , i − z | , , i )(1 + qz )(1 + q z ) , | , , i 7→ u ( − q + z + 2 qz + 2 q z + q z − qz )( u | , , i + u | , , i ) r (1 + qz )(1 + q z )+ q (1 + q ) u (1 − z )( | , , i − z | , , i )(1 + qz )(1 + q z ) , | , , i 7→ − q (1 + q ) u (1 − z ) | , , i (1 + qz )(1 + q z ) + (1 + q )(1 + q ) | , , i (1 + qz )(1 + q z )+ q (1 − z )(1 − qz ) | , , i (1 + qz )(1 + q z ) − q (1 + q ) u (1 − z ) | , , i (1 + qz )(1 + q z ) , | , , i 7→ (1 + q ) z | , , i qz + q (1 − z ) | , , i qz , | , , i 7→ − q (1 − z ) | , , i qz + (1 + q ) | , , i qz . Acknowledgments
The author thanks Ivan Cherednik, Yasushi Komori, Masato Okado and Yasuhiko Yamada for communi-cations and comments. This work is supported by Grants-in-Aid for Scientific Research No. 18H01141 fromJSPS. REFLECTION EQUATION 19
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