Matrix product solutions to the $G_2$ reflection equation

We study the $G_2$ reflection equation for the three particles in $1+1$ dimension that undergo a special scattering/reflections described by the Pappus theorem. It is a sixth order equation and serves as a natural $G_2$ analogue of the Yang-Baxter and the reflection equations corresponding to the cubic and the quartic Coxeter relations of type $A$ and $BC$, respectively. We construct matrix product solutions to the $G_2$ reflection equation by exploiting a connection to the representation theory of the quantized coordinate ring $A_q(G_2)$.

Here α 1 , α 2 are the simple roots and ∆ + denotes the set of positive roots which formally correspond to the spectral parameters. They are so ordered that the kth one from the left is s i1 · · · s i k−1 (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 2 case of the form R 12 (α 1 )G 132 (α 1 + α 2 )R 23 (2α 1 + 3α 2 )G 213 (α 1 + 2α 2 )R 31 (α 1 + 3α 2 )G 321 (α 2 ) = G 231 (α 2 )R 13 (α 1 + 3α 2 )G 123 (α 1 + 2α 2 )R 32 (2α 1 + 3α 2 )G 312 (α 1 + α 2 )R 21 (α 1 ), (2) where R is a solution to the Yang-Baxter equation by itself and G ∈ End(V ⊗ V ⊗ V) is the characteristic operator in the G 2 theory. It is a Yang-Baxterization of the G 2 Coxeter relation s 1 s 2 s 1 s 2 s 1 s 2 = s 2 s 1 s 2 s 1 s 2 s 1 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 2 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 1 . The equation (2) for generic symbols R and G without assuming a tensor structure on their representation space (i.e. without indices) has appeared as a defining relation of the root algebra of type G 2 [5,Sec.2]. In this paper we call (2) the G 2 reflection equation for simplicity.
The purpose of this paper is to construct families of solutions to the G 2 reflection equation with V = (C 2 ) ⊗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 of it is to embark on a quantization or a 3D version of the G 2 reflection equation. We introduce the quantized G 2 reflection equation which is a G 2 reflection equation (without spectral parameters) up to conjugation by a certain operator F acting on an auxiliary q-boson Fock space. Our finding (Theorem 4.1) is that with a suitable choice of the quantized scattering amplitude L and J, (3) coincides exactly with the intertwining relation [15, eq.(28)] of the A q (G 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 2 reflection equation (3). It offers a bonus; the equation/solution can be concatenated along the q-boson Fock space for arbitrary n times. The piled n layers of the 1 + 1 dimensional scattering diagrams can be viewed as a 3D lattice system in which adjacent layers may be interchanged locally according to (3) without changing the total statistical weight, a feature roughly referred to as 3D integrability. Anyway, to the n-concatenation of the quantized G 2 reflection equation, one can insert the spectral parameters and evaluate the intertwiner F away appropriately. It brings us back to the original G 2 reflection equation, thereby producing a solution to it for each n. Actually there are two such recipes called trace reduction and boundary vector reduction. They lead to the solutions (R tr (z), G tr (z)) and (R bv(z) , G bv (z)) 2 , respectively. By the construction they possess the matrix product structure containing n-product of L's or J's 3 where the trace and the sandwich ξ|(· · · )|ξ are taken over a q-boson Fock space. The detail will be explained in later sections. The solutions are trigonometric in the spectral parameter 4 . In fact R tr (z) and R bv (z) turn out to be the quantum R matrices [7,9] for the antisymmetric tensor representations of U p (A (1) n−1 ) and the spin representation of U p (D (2) n+1 ) with p 2 = −q −3 . 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 the Yang-Baxter and the reflection equations, respectively. To explore applications of the G 2 reflection equation is a future problem. For instance to architect commuting transfer matrices based on the G 2 reflection is an interesting issue.
The paper is organized as follows. In Section 2 we explain the interpretation of the G 2 reflection equation in terms of a special three particle scattering following [4,6]. The characteristic feature is the operator G which encodes the simultaneous reflection of one of the particles at the boundary and scattering of the other two. The world-lines of these particles form a configuration matching the classical Pappus theorem.
In Section 3 we formulate the quantized G 2 reflection equation by promoting R and G in (2) to the q-boson valued L and J. The L matrix (19) appeared first in [2]. The q-boson valued amplitude J (24)-(31) has been designed deliberately to validate Theorem 4.1. It does not split into the product of operators representing the single particle reflection and the two particle scattering. See (34).
In Section 4 after recalling basic facts on the representation theory of A q (G 2 ) [23], we state our key observation in Theorem 4.1. It identifies the quantized G 2 reflection equation with the intertwining relation between certain A q (G 2 ) modules.
In Section 5 we review the reduction of the tetrahedron equation (cf. [25]) to the Yang-Baxter equation following [2,17,16]. This construction has been illustrated in many literatures recently, e.g. [13,14], so we keep the 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 2 reflection equation. They lead to two families of solutions (R tr (z), G tr (z)) and (R bv (z), G bv (z)) to the G 2 reflection equation (2), where the latter is yet based on the conjectural relation (77). The role of the intertwiner F is curious. Although it is complicated and no closed formula is known, it does not give rise to a difficulty since the reduction procedure just eliminates it. Nevertheless F essentially controls the construction behind the scene in that it specifies precisely how the L and J are to be combined, how the spectral parameters should be arranged and 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 2 reflection equation and the intertwining relation (44) of the A q (G 2 ) modules. Appendix B contains explicit forms of (R tr (z), G tr (z)) and (R bv (z), G bv (z)) for small n.
2. G 2 reflection equation for three particle scattering 2.1. The G 2 reflection equation. Let V be a vector space and consider the operators depending on the spectral parameter z. We assume that R(z) satisfies the Yang-Baxter equation: 2.2. Scattering diagram; Pappus configuration. Let us describe the special three particle scattering related to the G 2 reflection equation. This is due to [4,6]. Consider the three particles 1,2,3 coming from A 1 ,A 2 ,A 3 and being reflected by the boundary at O 1 , O 2 , O 3 , respectively. See Figure 1. The bottom horizontal line is the 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 at a time is described by an element in V ⊗ V ⊗ V. Figure 1. Scattering diagram for the RHS of (7).
One can arrange the three particle world-lines so that the two particle scattering P 1 , P 2 , P 3 happen exactly at the same instant as the boundary reflection O 1 , O 2 , O 3 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 2 , O 3 and the intersection P 1 (and Q 1 ) and its projection O 1 onto the boundary. Let P 2 , P 3 be the points on the world-lines of particle 3 and 2 whose projection are O 2 and O 3 , respectively. In order to be able to draw the 5 Although these expansions do not specify r world-line for the last particle 1, the three points P 2 , P 3 and O 1 must be collinear. This is guaranteed by a special case of the Pappus theorem from the 4th century.
One can state it more symmetrically just by starting from P 1 , P 2 and their projection O 1 , O 2 onto the boundary. Let P ′ 1 , P ′ 2 be the mirror image of P 1 , P 2 with respect to the boundary. Then the three intersections P 1 O 2 ∩ O 1 P 2 , P 1 P ′ 2 ∩ P ′ 1 P 2 and O 1 P ′ 2 ∩ P ′ 1 O 2 are collinear; in fact they are P 3 , O 3 and the mirror image of P 3 . Let us call the so arranged scattering diagram a Pappus configuration. The reflection at O i with the simultaneous two particle scattering at P i will be referred to as a special three particle event (i = 1, 2, 3). Up to a translation in the horizontal direction, a Pappus configuration is parameterized by three real numbers. For instance one can specify it by the length of the segment O 1 O 2 and the (dual) reflection angles Then it is elementary to see We formally consider the infinitesimal angles hence replace (10) by w = u + v. In such a treatment, a Pappus configuration is labeled only by the two angles u and v. By a further substitution u = α 1 + α 2 and v = α 2 , (11) becomes Regard the symbols α 1 , α 2 formally as the simple roots of G 2 . They are transformed by the simple reflections s 1 , s 2 of the Weyl group as Thus we find and {θ 1 , . . . , θ 6 } yields the set of the positive roots of G 2 . The RHS of the G 2 reflection equation (7) is obtained by attaching R(e θ k ) to the two particle scattering at Q i and G(e θ k ) to the special three particle event at P i O i if it is the kth event starting from the left in Figure  1. Setting e u = x and e v = y, the assignment reads R 21 (x) : two particle scattering at Q 3 , G 312 (xy) : special three particle event at P 2 O 2 , R 32 (x 2 y 3 ) : two particle scattering at Q 1 , G 123 (xy 2 ) : special three particle event at P 3 O 3 , R 13 (xy 3 ) : two particle scattering at Q 2 , G 231 (y) : special three particle event at P 1 O 1 .
The indices for each operator correspond to the ordering of the relevant particles before the process. For instance just before the special three particle event at P 2 O 2 , the incoming particles are 3,1,2 from the top to the bottom, which is encoded in G 312 (xy). The LHS of the G 2 reflection equation (7) represents the Pappus configuration in which the time ordering of the processes are reversed. See Figure 2 3. Quantized G 2 reflection equation 3.1. q-bosons. Let F q = m≥0 C|m and F * q = m≥0 C m| be the Fock space and its dual equipped with the inner product m|m ′ = (q 2 ) m δ m,m ′ . We define the q-boson operators a + , a − , k on them by 2 ✲ 3 ✲ 1 ✲ Figure 2. Scattering diagram for the LHS of (7).
They satisfy ( m|X)|m ′ = m|(X|m ′ ). Let F q 3 , F * q 3 and A + , A − , K denote the same objects with q replaced by q 3 . Namely, The inner product in F q 3 is given by m|m ′ = (q 6 ) m δ m,m ′ differing from the F q case. However we write the base vectors as m|, |m either for F * q 3 , F q 3 or F * q , F q since their distinction will always be evident from the context. Note the q-boson commutation relations We will also use the number operator h defined by either for F q or F q 3 . One may regard k = q h+ 1 2 and K = q 3h+ 3 2 . The extra 1/2 in the spectrum of log q k is the zero point energy, which simplifies many forthcoming formulas.
This should not be confused with V in (5). In fact they will be related as V = V ⊗n later. (See around (53).) We introduce the q-boson valued L matrix by We attach a diagram to each component L γ,δ α,β ∈ End(F q 3 ) as follows 6 : The other configurations are to be understood as zero. So L may be regarded as defining a q-boson valued six vertex model in which the latter relation of (16) plays the role of "free-fermion" condition. See eq. (10.16.5)| d=0 in [1]. Explicitly we have Note the obvious properties 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 a characteristic feature of the G 2 scattering. It is defined by Each component J λ,µ,ν α,β,γ ∈ End(F q ) is depicted by a 90 • -degrees rotated special three particle event 7 We choose the operator J λ,µ,ν α,β,γ ∈ End(F q ) concretely as follows: Here u 1 , u 2 , u 3 , u 4 are parameters satisfying The operator s ∈ End(F q ) is defined by All the J λ,µ,ν α,β,γ 's not contained in the above list is zero. The weight conservation properties analogous to (20) and (21) hold:.
As an illustration we have The three particle diagram reduces to a direct product of two particle scattering and one particle boundary reflection if the dotted line were absent. Although it is not the case, the operator J almost splits into such a product as for some constants c, d. Here L denotes (19)| A ± →a ± ,K→k and K ν γ are the q-boson valued K matrix introduced in [16, eq.(9)].
3.4. Quantized G 2 reflection equation. Given L and J in Section 3.2 and 3.3, consider the G 2 reflection equation (7)| R→L,G→J that holds up to conjugation by an element This is an equality of linear operators on We fix the normalization of F by To explain the notation in (35) Practically, one can realize these operators from (18) and (22) by putting L γ,δ α,β and J λ,µ,ν α,β,γ at appropriate tensor components with a suitable permutations of the indices α, β, . . .. The equation (35) or equivalently (36) is a q-boson valued G 2 reflection equation without a spectral parameter up to conjugation. We call them the quantized G 2 reflection equation in analogy with the quantized reflection equation proposed in [16] for C 2 . It is depicted as follows. Here the indices 1,2,3 label the reflecting lines while 4,5,6,7,8,9 are attached to the scattering/reflection events. The latter group of indices are associated with the Fock spaces, and the q-bosons are acting on them in the direction perpendicular to this planar diagram. If one introduces such q-boson arrows going from the back to the front of the diagram, the operator F 456789 in the LHS (resp. RHS) corresponds to a vertex where the six arrows 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 depicted in the above.
The quantized G 2 reflection equation (35) is the set of 2 6 = 64 equations like (39) corresponding to the choice of a, b, c, i, j, k ∈ {0, 1} in (38). In the next section they will be identified with the intertwining relation of certain A q (G 2 ) modules. 4. A q (G 2 ) and its intertwiner 4.1. Intertwining relation of A q (G 2 ) modules. The quantized coordinate ring A q (G 2 ) is a Hopf algebra which can be realized by 49 generators (t i,j ) 1≤i,j≤7 obeying the so-called RT T type quadratic relations and some additional ones. They are available in [20], which was adopted in [15,Sec.3.3.3] in the form directly relevant to this paper. Their concrete form is not necessary here. What we need is the two fundamental representations π i : A q (G 2 ) → End(F qi ) associated with the simple reflections s i (i = 1, 2), where q 1 = q, q 2 = q 3 .
Let Φ ∨ be the intertwiner. Namely it is the map F q ⊗F q 3 ⊗F q ⊗F q 3 ⊗F q ⊗F q 3 → F q 3 ⊗F q ⊗F q 3 ⊗F q ⊗F q 3 ⊗F q characterized by π 212121 Φ ∨ = Φ ∨ π 121212 up to normalization. Set Φ = Φ ∨ • P where P is a linear map reversing the order of the six-fold tensor product as P (x 1 ⊗ x 2 ⊗ · · · ⊗ x 6 ) = x 6 ⊗ x 5 · · · ⊗ x 1 . Thus there exists the unique Φ such that π 212121 (g) Φ = Φ π ′ 121212 (g) ∀g ∈ A q (G 2 ) (π ′ 121212 := P π 121212 P ), 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 where the sums are taken over 1 ≤ l 2 , . . . , l 6 ≤ 7. In this way the intertwining relation (42) boils down to the 49 equations (44). Although the lists of π 1 (t i,j ), π 2 (t i,j ) in (40) are pretty sparse, some equations become lengthy including typically 16 terms on one side or both. We do not display them all here but present a few examples.
(45) g = t 2,6 :  Henceforth we shall identify F and Φ and write F to also mean the intertwiner Φ. Let us quote some basic properties of F from [15,Sec.4.4]. Set Then the following properties are valid: 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 where x and y are free parameters. We let F also act on ω| ∈ F * It is possible to make a tedious computer program to calculate F abcdef ijklmn for any given indices by using (44). However unlike the A q (A 2 ) and A q (C 2 ) cases, an explicit general formula is yet to be constructed. At q = 0 F abcdef ijklmn 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 100102 . L 124 L 135 L 236 R 456 = R 456 L 236 L 135 L 124 ∈ End( This is a Yang-Baxter equation up to conjugation by R ∈ End(F q 3 ⊗ F q 3 ⊗ F q 3 ). Such an R is unique up to overall normalization and is known to satisfy the tetrahedron equation R 124 R 135 R 236 R 456 = R 456 R 236 R 135 R 124 among themselves. See [10] for the approach from the representation theory of the quantized coordinate ring A q (A 2 ), [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 3 ) ⊗3 by ( a| ⊗ b| ⊗ c|)R (|i ⊗ |j ⊗ |k ) = ( a| ⊗ b| ⊗ c|) R(|i ⊗ |j ⊗ |k ) . In this paper we will only need the following properties where x, y are free parameters. The relation (51) was proved in [17, Pr.4.1] 9 .
Alternatively one may sandwich (55) between the bra vector ( where a is a dummy label for the auxiliary Fock space a F q 3 . The normalization factors ̺ tr (z) and ̺ br (z) will be specified in (68). Now (56) and (57) are both stated as the Yang-Baxter equation 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 the Yang-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 more general ones in [17]. The solutions R tr (z) and R bv (z) have been identified with the quantum R matrices for the antisymmetric tensor representations of U p (A

Reduction of quantized G 2 reflection equation
Starting from the quantized G 2 reflection equation (36), one can perform two kinds of reductions similar to Section 5 to construct solutions to the G 2 reflection equation (7) in the matrix product form. This is the main result of the paper which we are going to present in this section.
(80) 11 The two relations in (77) are actually equivalent due to the right property in (47).
The R and G matrices are linear operators on V ⊗ V and V ⊗ V ⊗ V with V ≃ (C 2 ) ⊗n and trigonometric in the spectral parameter. The special three particle event characteristic to the G 2 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 the antisymmetric tensor representations of U p (A (1) n−1 ) and the spin representations of U p (D (2) n+1 ) with p 2 = −q −3 .