Triangular solutions to the reﬂection equation for U q ( c sl n

. We study solutions of the reﬂection equation related to the quantum aﬃne algebra U q ( c sl n ). First, we explain how to construct a family of stochastic integrable vertex models with ﬁxed boundary conditions. Then, we construct upper-and lower-triangular solutions of the reﬂection equation related to symmetric tensor representations of U q ( c sl n ) with arbitrary spin. We also prove the star-star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained.

Corwin and Petrov in [11] (see also [12] for a review) showed that many currently known integrable 1+1-dim KPZ models come from suitable limits of stochastic higherspin vertex models.Originally, they studied these higher-spin models on the line (see also [13,14]).However, later, this was generalized to half-line, half-quadrant processes with open and special fixed boundary conditions (see, for example, [12,[15][16][17]).
Sklyanin [18] developed a general theory of quantum integrable systems with boundaries based on the so-called reflection equation.The reflection equation first appeared in a 2-dim factorised scattering theory on a half-line [19], with the scattering matrix satisfying the Yang-Baxter equation [20][21][22][23].
In this paper, we construct special triangular solutions of the reflection equation for the higher-spin stochastic vertex models related to symmetric tensor representations of U q ( sl n ).These solutions generalise some of our results from [24].Let us also mention a recent paper [25], where the author constructed a special oneparametric family of integrable open boundaries for the so-called q-Hahn processes, related to U q ( sl 2 ).
It is important to note that we work with a stochastic R-matrix, which is a twisted version [26] of the standard U q ( sl n ) R-matrix.As was noticed in [27], this leads to a factorised L-operator, which may be why all our solutions possess factorised matrix elements.
The paper is organised as follows.In Section 2 we review the theory of boundary integrable systems and show how to construct stochastic versions of the transfer-matrix and hamiltonian.In Section 3 we introduce required notations, define the stochastic U q ( sl n ) R-matrix and discuss its properties.We also introduce two higher-spin Loperators required later in the text.In Section 4 we prove the star-star relation from [28], which we use later.Sections 5 and 6 contain the main results of the paper.Namely, we construct left and right boundary solutions for the reflection equation in (5.7), (5.16), (6.19) and (6.20) and prove their properties.In Section 7 we give the proof of the reflection equation for a generalized model presented in [24].In Conclusion, we discuss the results obtained and unsolved problems.Finally, in Appendices, we prove some technical results used in the text.

Reflection equation and commuting transfer-matrices
We start with the Yang-Baxter equation whose solutions R 12 (x) ∈ End (V 1 ⊗ V 2 ) act non-trivially in the tensor product of two vector spaces and describe the behaviour of the system in the bulk.We assume that the R-matrix satisfies the following properties: • R t2 12 (x) is non-degenerate, i.e. one can correctly define the matrix where t i denotes transposition with respect to V i , i.e. in the matrix components it has the form R t2 12 (x) where we assume that V 1 ∼ = V 2 ≡ V .Here P is a permutation operator, acting in a usual way where f (x) is a scalar function satisfying f (x) = f (x −1 ).In fact, one can check that if V 1 ∼ = V 2 , then the last property directly follows from the regularity condition (2.3) and the Yang-Baxter equation (2.1) (see, for example, [29]).
• Crossing unitarity condition where r ∈ C \ {0}, and M ∈ End(V ) is a constant matrix.Note that the crossing unitarity is a stronger condition than (2.2) [24,30], since the expression directly follows from (2.5).In particular, if we consider the R-matrix as an intertwiner of finite-dimensional U q ( g) modules, then it follows from the quantum group arguments that conditions of unitarity (2.4) and crossing unitarity (2.5) are satisfied [31,32].In this instance, M = q −2ρ , r = q h ∨ , where ρ is a half-sum of positive roots, and h ∨ is a dual Coxeter number of the Lie algebra g.In what follows, we will focus on g = sl n (C) case, and as we will see in the next chapter, the matrix M is diagonal and satisfies an additional property Now we come to describing the boundaries: for a given solution to the Yang-Baxter equation (2.1), we define the so-called reflection equation and the dual reflection equation where the matrix R 12 (x) is defined in (2.2).The solutions K i (x) , K i (x) act nontrivially on V i (i = 1, 2) and describe the right and left boundaries, respectively.Knowing the solution to the reflection equation (2.8) and using the commutativity property (2.7), the solution to the dual reflection equation (2.9) can be expressed in the form (2.10) Following [18,33], we can also consider another reflection equation related to the inverse R-matrix R 21 x −1 ∼ = R −1 12 (x): It is stated that if K(x) is a solution to equation (2.11), then the expression solves the dual reflection equation (2.9).The proof of this fact is quite technical, and we provide it in Appendix A for the reader's convenience.The mapping (2.12) defines an isomorphism between the solutions to the dual reflection equation (2.9) and the equation (2.11) (see [18] for this and other isomorphisms): the inverse mapping has the form which we also prove in Appendix A. Substituting y = 1 into the reflection equation (2.8), we get which should be true for any x.Apparently, the only reasonable solution to (2.14) is K(1) ∼ I.In this paper we shall always assume the normalization Solutions to the reflection equation (2.8) and the dual reflection equation (2.9) allow us to construct a double-row transfer matrix (2.16) Transfer-matrices (2.16) form a one-parameter family of mutually commuting quantities: [T (x), T (y)] = 0, (2.17) and this property implies the integrability of the model.The proof of (2.17) can be found in [18].Taking into account (2.3) and (2.15), we obtain Finally, we can calculate the Hamiltonian of the integrable system associated with the transfer-matrix (2.16) where The operators B L and B R act only in the first and the N -th component of the Hilbert space.Let us notice that the operator B L at the left boundary is proportional to the derivative of the matrix K(x) as can be easily seen from differentiating (2.13).Now let us discuss under what conditions the Hamiltonian and the transfer-matrix define stochastic processes.
We say that an (2.28) Dividing T (x) by a scalar factor c(x), we obtain the condition (2.22).If the rank of H is smaller than dim(V ) N − 1, we cannot say anything about the stochasticity of T (x).Indeed, in the case of periodic boundary conditions the hamiltonian may have additional zero modes and the transfer matrix is not stochastic in general.However, in all cases with fixed boundary conditions, which we considered, the rank of the Hamiltonian H was dim(V ) N − 1 and this implies (2.28).
Let us also notice that we don't know how to show the stochasticity of the transfermatrix (2.16) directly from (2.24).The main difficulty is that T (x) is defined in terms of K(x) related to K(x) via (2.12).
3. Stochastic R-matrix for U q ( sl n ) In this section we introduce the stochastic R-matrix related to the symmetric tensor representations of the quantum affine algebra U q ( sl n ) and consider its basic properties.
First, we start by introducing index notations.We denote by i = {i 1 , . . ., i m }, the m-component set of non-negative integers i k ∈ Z + , k = 1, . . .m.We also define some operations on these sets: where a, b ∈ Z m + .The permutation τ is defined as We will also need one more operation on indices: for a given J ∈ N and i ∈ Z m + , |i| ≤ J, we introduce the mapping For a positive integer J ∈ N and n ≥ 2, we define a finite-dimensional vector space V (n) J with basis vectors having the form (do not confuse this notation with the row vectors from the previous section).The dimension of the space We will also use an equivalent notation for the basis vectors of V (n) where we extended the vector |i by adding an additional component i n = J − |i|.
The introduced vector space naturally inherits the structure of finite-dimensional symmetric tensor representations of the quantum affine algebra U q ( sl n ) [14].For further purposes, it will be essential for us to consider a realization of the symmetric tensor representation of sl n (C) Lie algebra on V J : ) where e i , f i and h i (i = 1, . . .n − 1) are the standard sl n (C) generators, | ᾱ = |α 1 , α 2 , . . ., α n as defined in (3.5) and e j represents the vector with 1 at the j-th place and 0 otherwise.For what follows, we will need to find an expression for the half-sum of positive roots ρ of sl n (C), and a simple analysis shows that it has the form Now we discuss the stochastic R-matrix related to symmetric tensor representations, see [14,27] for more details.For arbitrary positive integers I, J ∈ N we introduce a linear operator S I,J (x) ∈ End(V J ), whose action on the basis i ′ ⊗ j ′ (in terms of (3.4)) has the form The matrix elements can be written in a compact way as where we defined the function Here we use the standard notation for q-Pochhammer symbols and q-binomial coefficients m k q def = (q; q) m (q; q) k (q; q) m−k . (3.12) Let us notice that the original parameter q of the quantum group U q ( sl n ) enters the expression for the stochastic R-matrix only via q 2 as indicated by the subscript of the function Φ q 2 in (3.9).Such notations were introduced in [14] and we prefer to follow them.
The introduced R-matrix was constructed in [27] from the corresponding 3D Roperators, where it was shown that it satisfies the Yang-Baxter equation defined on We can also obtain different symmetries of the R-matrix (3.9) from the symmetries of corresponding 3D R-matrix.Before we describe them, we define a symmetric version of the R-matrix (3.9) RI,J (x) One can check that in the case n = 2 and I = J = 1, the R-matrix (3.9) corresponds to the stochastic symmetric 6-vertex model It will be also convenient to represent the function Φ from (3.10) in another form where the function V x (a, b) is defined as (q 2 ; q 2 ) ai−bi . (3.17) This function was introduced in [28] in the formulation of Ising-type model related to the U q ( sl n ) algebra, and the use of it will allow us to write down a number of relations below in a more compact form.
Then we can express the symmetries, proven in [27] from the properties of the 3D R-matrix, in terms of the symmetric R-matrix (3.14) and the function (3.17) as follows RI,J (x) RI,J (x) where the indices are related by the permutation τ defined in (3.2).Here we introduced an additional operation on indices We also have another symmetry based on the transformation σ from (3.3): Now let us recall the other basic properties of the stochastic R-matrix (3.9): This identity is based on the following summation rule: which will be important for us in describing stochastic processes.The proof of this fact can be done by induction on n and using the q-Vandermonde summation formula from Appendix B (see [14] for details).As a consequence of the summation formula (3.23), we directly obtain the stochastic condition (3.22).
• Regularity S I,I (1) = P 12 , ( where P is a permutation operator, acting on I .This property can be easily checked from the definition of the R-matrix (3.9).
• Unitarity condition S I,J (x)S J,I x −1 = I ⊗ I. (3.25) • Crossing unitarity condition where M is the diagonal matrix and the scalar function in the right-hand side is x 2 q 2−I−J ; q 2 I x 2 q 2n−I+J ; q 2 I (x 2 q 2−I+J ; q 2 ) I (x 2 q 2n−I−J ; q 2 ) I . (3.28) To verify that the scalar factor in the unitarity condition (3.25) is indeed equal to 1, one needs to apply the stochastic property (3.22).As mentioned in the previous section, (3.26) follows from the general theory of quantum groups.Using the formula (3.7) and the fact that the dual Coxeter number h ∨ = n for the Lie algebra sl n (C), we come to the expression in the left-hand side of (3.26).
It remains to show that the function in the right-hand side of the crossing unitarity relation is given by (3.28).It is enough to check the relation for any diagonal element, say, for i = j = i ′′ = j ′′ = 0 (i ′ and j ′ are internal summation indices).Note that the R-matrix contains an internal summation (see definition (3.9)).However, we can remove internal sums by applying the symmetry (3.18) to both factors in (3.29), since in this case the R-matrix contains only one term.
Then one can rewrite (3.29) using (3.16), and we come to the formula This identity is a consequence of the equation (6.16), that will be considered in the following sections.And this ends the proof of the crossing unitarity relation.

The star-star relation
Using any of the symmetries (3.18), (3.19), one can obtain the star-star relation for U q ( sl n ), written in terms of the functions (3.17): which was presented by Bazhanov and Sergeev in [28] as a conjecture.Namely, let us introduce the new matrix indices which automatically imply the conservation law i + j = i ′ + j ′ coming from the Kronecker delta-symbol in the definition of the R-matrix (3.9).Now let us note the property of the function Φ with permuted indices and the property of the function which can be easily verified from the corresponding definitions (3.10), (3.17

Stochastic right boundaries
In this section we construct special cases of the right integrable boundaries, which are the upper-triangular solutions to the reflection equation and those that are generated from the former by applying symmetries of the stochastic R-matrix.
The reflection equation for the stochastic R-matrix (3.9) has the form S I,J x y K I (x) S J,I (xy) K J (y) = K J (y) S I,J (xy) K I (x) S J,I x y . (5.1) First, we show that for a solution K J (y) of (5.1), we can construct another solution where µ ∈ C is an arbitrary parameter.To prove this statement, we will write the reflection equation (5.1) for K J (y) in index notation, but with both external and internal summation indices replaced by j → σ −1 j, defined in (3.3), J (y) J (y) I (x) 3) Now let us note that the R-matrix (3.9) possesses another symmetry which follows from the relations (3.18) and (3.21).Applying (5.4) and (3.18) to (5.3) and using the fact that one can check that all pre-factors on both sides of (5.3) cancel, and we come to the reflection equation for K J (y).Using the symmetry (5.2) k times, k = 1, . . ., n−1, we will obtain a set of different solutions to the reflection equation (5.1).Moreover, we can adjust the free parameter µ ∈ C in (5.2) at each step to satisfy the stochastic condition.
For example, the solutions K (2,3) (x) from [34] can be obtained from K (1) (x) by using the symmetry (5.2) for n = 3.However, the solution K (4) can not be obtained in this way.Now let us construct the upper-triangular boundary matrices.We will use the same approach as described in [24] for the U q ( sl 2 ) case.Namely, we will start from considering the reflection equation for I = J = 1 and show that it admits the solution K 1 (y) l j = Φ q 2 j|l; which is a generalization for U q ( sl n ) with arbitrary n.Here ν ∈ C is a free parameter.Then, using this solution, we will construct a system of linear recurrence equations for I = 1 and arbitrary J. And, at the final step, we will show that an upper-triangular stochastic solution to these equations has the form K J (y) l j = Φ q 2 j|l; which satisfies the stochastic condition as can be seen from the formula (3.23).
It is convenient to consider the reflection equation (5.1) in a matrix form, that is, to use the expressions for L-operators (3.31) and (3.32).In a particular case J = 1, we come to a simpler formula x y where the summation goes over internal repeated indices, and the upper-triangular matrix K 1 (x) has the form It is clear from the formulas (5.9) and (5.11) that the reflection equation (5.10) has the same form for any n: indeed, it only depends on the external indices e α , e β , e α ′′′ , e β ′′′ (more precisely, on the relative order of the numbers α, β, α ′′′ , β ′′′ ), and each factor in the relation gives no more than 2 nontrivial terms.This means that it is enough to check the correctness of the equation for, say, n = 5 (to consider all possibilities for index permutations).And, indeed, the explicit calculation shows that the equation is valid in this case, which proves (5.6).Now, based on the solution for I = 1, we can consider the reflection equation for an arbitrary parameter J where K 1 (x) is given by (5.6).Then, substituting (3.31), (3.32) in (5.12) in matrix form and performing some cumbersome calculations, we obtain a system of n 2 linear equations, that are polynomial in x and y.Decoupling with respect to x, we get a set of recurrence relations for K J (y) l j .A careful analysis of these equations shows that there are n linearly independent ones: where we assume that [K J (y)] l j = 0 unless j α ≥ 0, l α ≥ 0, |j| ≤ J and |l| ≤ J.And in the last step one can check that (5.7) solves these recurrence equations.Thus we constructed the integrable upper-triangular right boundary solution.
Applying (5.2) to (5.7) and choosing µ = ν, we obtain a stochastic lowertriangular solution of the form (5.16) where the operation on indices σ is defined in (3.3), and we introduced a new notation Φ q (j | l; x, y) = q Q(j,l)−Q(l,j) Φ q (j | l; x, y) . (5.17) In the second line in (5.16) we applied the relation.
The stochastic condition (5.8) for the lower-triangular solution follows from the identity where we used the relations (5.18), (4.3) and the summation rule (3.23).
It is interesting to compare the obtained solutions (5.7) and (5.16) with already known results.In [35] the author introduced a solution to the reflection equation related to the stochastic R-matrix .20)derived in [36] through the fusion procedure.The solution from [35] has a similar structure to (5.7) and also includes the function Φ as a building block.
In [37] a full classification of the right boundaries for the case J = 1 was given.It is based on the classification with four particles of two special types (four positive integers): slow (s 1 , s 2 ) and fast (f 1 , f 2 ) ones, which satisfy the constraints (5.21) Note that the authors of [37] use different index notations, starting from 1 rather than 0. Each solution depends on two real parameters α, γ with the choice of two boundary matrices B 0 (α, γ|s 1 , s 2 , f 2 , f 1 ) and B (α, γ|s 1 , s 2 , f 2 , f 1 ) .Then one can check that the solution (5.7) for J = 1 corresponds to the case of the matrix B (α, γ|s 1 , s 2 , f 2 , f 1 ) with the slow s 1 = s 2 = 1 and fast f 1 = f 2 = n species and the parameters and the lower-triangular solution (5.16) corresponds to the parameters To illustrate the structure of the right boundary matrices (5.7), (5.16), we listed them in Appendix D for a particular case n = 3 and J = 2.

Stochastic left boundaries
Now we will construct the left upper-and lower-triangular boundaries using the results from the previous sections.The starting point for this calculation is the dual reflection equation where as usual First, let us notice that the matrix M (3.27) satisfies the compatibility condition (2.7): [M ⊗ M, S I,J (x)] = 0. ( As discussed in Section 2, we can find the solution to (6.1) in terms of K I (x) from (5.7) However, we are also interested in solutions KI (x) of the reflection equation (2.11) which enter the definition of the hamiltonian rates, see (2.20).The corresponding reflection equation takes the form: First, let us discuss the symmetries of this equation.If K(1) J (y) is a solution of (6.5), then is also a solution.This can be shown by using the symmetry (5.4) of the R-matrix and performing a calculation similar to (5.3).
Proceeding in a similar way, one can show that if K J (y) solves (5.1), then KJ (y) l j = K J (y −1 ) τ σl τ σj (6.7) solves (6.5).Let us emphasize that the formula (6.7) is a consequence of the symmetry (5.4).However, we also have a general construction for the matrix K(x) given by (2.13) and (2.10).In the rest of this section we will show that it gives the same result (6.7) up to the transformation (6.6).
Combining (2.13) with (6.4), we obtain where we assume that I = J to correctly define the permutation operator acting on two isomorphic spaces.We start our analysis with the upper-triangular solutions (5.7).Substituting (3.27) in (6.8) and replacing an arbitrary parameter ν → 1 νq n , we obtain the following formula: We claim that (6.9) can be transformed to the following form ) where in the last line we used (5.18) and introduced the function Combining the first line in (6.10) with the symmetry transformation (6.6), we see that the formula (6.10) is equivalent to (6.7) up to a scalar factor (6.11).
To prove the equivalence of (6.9) and (6.10), we want to apply certain summation formulas.A simple analysis shows that, by using a symmetry transformation (6.12) in (6.9), we obtain the terminating series, truncated in a natural way by J.
Rewriting all functions Φ in terms of V 's and canceling out the pre-factors, we obtain that the equality of (6.9) and (6.10) is equivalent to the proof of the relation d, m) .
(6.13)Here we introduced matrix indices In these notations, all indices satisfy the constraints Two summations in (6.13) can be done by using the summation formulas where and The relation (6.16) is proven in Appendix C. The relation (6.18) easily follows from the star-star relation (4.1).Indeed, let us substitute b = a in (4.1).Then the sum in the LHS disappears, since a ≥ m and m ≥ b = a and (4.1) reduces to (6.18).Since the solution to the reflection equation is defined up to an arbitrary factor, we divide (6.10) by λ (n) J (x) and obtain for the integrable left upper-triangular boundary KJ (y) l j = Φ q 2 l − j|l; y 4 , y 2 νq J , ( which satisfies the stochastic property due to (3.23).Now let us construct the stochastic lower-triangular left boundary solutions.They are obtained by applying the symmetry (6.6) with µ = ν to the first line of (6.10) and removing the scalar factor λ (n) J (x).As a result, we have KJ (y) l j = Φ q 2 σj | σl; where Φ q is defined in (5.17).
In Appendix D we listed left boundary matrices (6.19), (6.20) for a particular case n = 3 and J = 2.

Another model
In this section we will not impose the requirement I, J ∈ N, i.e the R-matrix (3.9) acts in the tensor product of infinite dimensional modules with basis vectors having the form The R-matrix (3.9) degenerates when we choose the spectral parameter as x = q (J−I)/2 [14,27]: which suggests to define a new stochastic non-difference type R-matrix Now we consider x, y ∈ C as spectral parameters and the Yang-Baxter equation takes the form S 12 (x, y) S 13 (x, z) S 23 (y, z) = S 23 (y, z) S 13 (x, z) S 12 (x, y) , (7.4) which directly follows from (3.13).It is easy to see that the internal sums in (7.4) contain only a finite number of terms, so there is no problem with convergence.For a given solution to the Yang-Baxter equation (7.3), we define the reflection equation of a non-difference type In [24] it was suggested that this equation admits an upper-triangular stochastic solution of the form where z ∈ C is an arbitrary parameter and there is no constraint between spectral parameters x and x (see the equation (5.10) from [24]).
Here we give the proof of this fact suggested in [38], which is based on the star-star relation (4.1) and an orthogonality property: The last equality directly follows from the relation (B.6) and the q-Vandermonde summation formula (B.2).We start the proof by writing the reflection equation (7.5) in index notation ) where, as usual, the summation goes over repeated indices.Then we introduce new external indices summation indices for the left-hand side and for the right-hand side After canceling out similar factor on both sides, the reflection equation in terms of the functions V x (a, b) takes the form V z x (f , d 1 ) .
(7.12)Then, sequentially applying the star-star relation (4.1) centered at d 1 and orthogonality formula (7.7) with respect to d 2 to both sides of the relation (7.12), we come to the conclusion that the left and right sides coincide: which, however, does not include an additional free parameter as in (7.6).

Conclusion
In this paper we constructed triangular solutions for the reflection equation corresponding to symmetric tensor representations of U q ( sl n ).It is quite remarkable that all constructed solutions are built from the same function Φ q (γ | β; λ, µ) defined in (3.10).As a result, both reflection and dual reflection equations (5.1), (6.1) reduce to multiple sum identities among the Φ-functions (or V -functions (3.17)).It would be natural to expect that (5.1), (6.1) can be proven by a repeated application of the star-star relation (4.1).However, we failed to find a correct sequence of the star-star relations which would prove these equations.This is the reason why we introduced the L-operators (3.31) and (3.32) to prove that the solution (5.7) solves the reflection equation (5.12).We are not aware of any other nontrivial identities among V -functions except (4.1) and believe that such an algebraic proof may still exist.We expect that general stochastic solutions for the U q ( sl n ) algebra should have one extra parameter as demonstrated in [24] for U q ( sl 2 ) and in [37] for U q ( sl n ) with J = 1.Since our triangular solutions are factorised, they may serve as building blocks for general solutions similar to the expression of the stochastic R-matrix (3.9).However, we have not yet found such a general representation and aim to do that in the future.which is valid for any matrices A and B. Then the following calculation proves (A.7): x 2 P 0 ′ 1 = tr 0,0 ′ K0 (x) R 00 ′ 1 x 2 P 00 ′ R 0 ′ 1 1 x 2 P 0 ′ 1 = tr 0 P 01 K1 (x) tr 0 ′ P 00 ′ R 10 1 x 2 R 10 ′ ).If we take(3.18)or (3.19) and use (3.16) together with (4.3) and (4.4), one can see that all pre-factors on both sides of (3.18) or (3.19) cancel out, and we come to the star-star relation (4.1).

( 5 . 9 )
The last expression allows us to analyze the structure of the reflection equation for I = J = 1.So we want to show that the right boundary (5.6) satisfies the relation e α ′ ,e β ′ ,e α ′′ ,e β ′′ S 1,1
The solutions(6.19),(6.20),related to the left boundary, have a similar structure: