Stochastic $R$ matrix for $U_q(A^{(1)}_n)$

We show that the quantum $R$ matrix for symmetric tensor representations of $U_q(A^{(1)}_n)$ satisfies the sum rule required for its stochastic interpretation under a suitable gauge. Its matrix elements at a special point of the spectral parameter are found to factorize into the form that naturally extends Povolotsky's local transition rate in the $q$-Hahn process for $n=1$. Based on these results we formulate new discrete and continuous time integrable Markov processes on a one-dimensional chain in terms of $n$ species of particles obeying asymmetric stochastic dynamics. Bethe ansatz eigenvalues of the Markov matrices are also given.


Introduction
Quantum groups and theory of quantum integrable systems provide efficient algebraic and analytic tools to evaluate non-equilibrium characteristics in stochastic processes in statistical mechanics. See for example [12,28,24,27,32,8,5,7,6] and references therein. Typically in such an approach, one sets up a row transfer matrix or its derivative as in usual vertex or spin chain models [1], and seeks the situation that admits an interpretation as a Markov matrix of a certain dynamical system on a one-dimensional chain. It leads to a postulate more stringent than the models in the equilibrium setting. Namely, the transfer matrix or its derivative must have non-negative off-diagonal elements and they should further satisfy a certain sum-to-unity or sum-to-zero conditions assuring the total probability conservation depending on whether the time evolution is discrete or continuous, respectively.
One may try to modify a given transfer matrix so as to fit them, but doing so indiscreetly leads to a loss of the essential merit, the integrability or put more practically, the Bethe ansatz solvability. In this way an important general question arises; Can one architect the transfer matrices or their constituent quantum R matrices so as to fulfill the basic axioms of Markov matrices without spoiling the integrability?
The aim of this paper is to answer it affirmatively for the R matrix associated with the symmetric tensor representations of the Drinfeld-Jimbo quantum affine algebra U q (A (1) n ) [9,14]. By now, the quantum R matrix itself is a well-known classic. Nevertheless investigation of the above question elucidates a number of remarkable insights which have hitherto escaped notice.
In this paper we establish the above formula for general n substantially in Theorem 2. Our strategy is to resort to the characterization of the R matrix as the commutant of U q (A (1) n ) [9,14], which takes advantage of the most essential machinery of the theory rather than manipulating concrete formulas as in the preceding works. Our proof of Theorem 6 also captures the sum-to-unity relations (17) conceptually from the representation theory of quantum groups. It manifests that the totality of those relations is nothing but the U q (A n )-orbit of the unit normalization condition (5) on the trivial highest weight vector. Such a mechanism is quite likely to work similarly in many other algebras and representations.
Based on these findings on the R matrices, we first formulate two kinds of commuting families of discrete time Markov processes on a one-dimensional chain. They are described in terms of n species of particles obeying totally asymmetric dynamics with and without constraint on their numbers occupying a site or hopping to the right at one time step. From the constraint-free case we then further extract the continuous time versions by differentiating the Markov transfer matrix by λ in Φ q (γ|β; λ, µ) parameterizing the commuting family. The procedure is analogous to the standard derivation of spin chain Hamiltonians as in [1, eq.(10.14.20)]. A curiosity encountered in our model is that the transfer matrix admits two "Hamiltonian points" λ = 1 and λ = µ at which such calculations can naturally be executed as in (60). They lead to the two Markov matrices H andĤ which are interpreted as n-species totally asymmetric zero range processes (TAZRPs) in which particles hop to the right and to the left adjacent site, respectively. By the construction the commutativity [H,Ĥ] = 0 holds, therefore the superposition aH + bĤ yields an integrable asymmetric zero range process in which n species of particles can hop to either direction.
In the TAZRP associated toĤ, the relevant transition rate (59) is similar to the above. In particular, at µ = 0 and ǫ = 1 it reduces to At q = 0, it gives rise to a kinematic constraint 1≤i<j≤n γ i (β j − γ j ) = 0 which is translated into a simple priority rule on the species of particles that are jumping out together. It precisely reproduces the n-species TAZRP explored in [17,18] under a suitable adjustment of conventions. Once the models are identified in the framework of quantum integrable systems, spectra of the Markov matrices with the periodic boundary condition follow from the Bethe ansatz. We present the eigenvalue formulas adjusted to the stochastic setting under consideration. Steady state eigenvalues, given explicitly in (77), are naturally identified with those associated with the trivial Baxter Q functions.
The layout of the paper is as follows. In Section 2 we derive several properties of the U q (A n ) quantum R matrix R(z) and its stochastic versions S(z) and S(λ, µ) that are essential for applications in the subsequent sections. In Section 3 the commuting transfer matrices built upon the S(z) and S(λ, µ) are shown to satisfy the basic axioms of Markov matrices in a certain range of parameters. The associated stochastic processes are formulated, which generalize various known models for n = 1. Section 4 presents the Bethe ansatz eigenvalue formulas of the Markov matrices together with some examples of steady states. Section 5 is a summary. Appendix A contains explicit forms of simple examples of the R matrix.
As a U q (A n )-module, the highest weight vector in V l is |0, . . . , 0, l , which is also annihilated by all the f i 's except f n . Thus V l is actually the l-fold symmetric tensor of the anti-vector representation which corresponds to the n × l rectangular Young diagram.
Let us further introduce R(z) = R l,m (z) = PŘ l,m (z) ∈ End(V l ⊗ V m ), where P (|α ⊗ |β ) = |β ⊗ |α is the transposition. The both R(z) andŘ(z) will be called the quantum R matrix or just R matrix for short. Its action is expressed as where α ∈ B l , β ∈ B m and the sums are taken over γ ∈ B l , δ ∈ B m . The matrix elements R(z) γ,δ α,β are rational functions in z and q. In principle, they are computable either by the fusion [16] from the (l, m) = (1, 1) case (bottom-up) or by taking the image of the universal R (top-down). Practically an efficient alternative is to evaluate the trace of the product of the three-dimensional R operators [15,3,4,20] satisfying the tetrahedron equation. This approach has been developed in [3,25,26,23,21,22] as an outgrowth of the pioneering works [33,2,29]. Examples in Appendix A have been generated by this method by using [22, eq.(2.24)]| ǫ1=···=ǫn=0 . See also [18] for a recent application of the tetrahedron equation to a multispecies TAZRP. We depict the matrix element of the R matrix as suppressing dependence on n, z, q, and also l, m associated with the horizontal and vertical lines, respectively. This picture matches the action ofŘ(z) in (6) viewed in the ր direction. The relation (4) with g = k i tells the weight conservation property that R(z) γ,δ α, The most significant property of the R matrix is the Yang-Baxter equation [1] which is presented in two equivalent forms: where the lower indices in (9) specify the components on which R(z) acts nontrivially 3 . The relations (8) and (9) hold as the operators The equality of the matrix element for |α ⊗ |β ⊗ |γ → |α ′′ ⊗ |β ′′ ⊗ |γ ′′ in (9) is depicted as The R matrix also satisfiesŘ The former is called the inversion relation. In the latter α ′ = (α n+1 , . . . , α 1 ) denotes the reverse array of α = (α 1 , . . . , α n+1 ) and β ′ , γ ′ , δ ′ are similarly defined. It is a corollary of Note that the q-binomial factors in (13) tell that R(q l−m ) γ,δ α,β = 0 unless β ≥ γ or equivalently α ≤ δ under the condition α + β = γ + δ. Here and in what follows, u ≥ v for u, v ∈ Z k for any k is defined by u − v ∈ Z k ≥0 and ≤ is defined similarly. The condition l ≤ m in the claim matches this property. It is interesting that the "inter-color coupling" enters only via ψ apparently. See the end of Appendix A for an example. For the proof we prepare Lemma 3. For any i ∈ Z/(n + 1)Z, the following equalities are valid:

Proof. A direct calculation.
Proof of Theorem 2. R(z) is not singular at z = q l−m . See for example [22, eq.(6.16)]. Thus it suffices to check that the RHS of (13) satisfies (4) and (5). The latter is obvious. The relation (4) with g = k i means the weight conservation and it holds due to the factor δ γ+δ α+β . In the sequel we show (4) for g = f i . The case g = e i can be verified similarly. Let the both sides of (4) act on |α ⊗ |β ∈ V l ⊗ V m and compare the coefficients of |δ ⊗ |γ in the output vector. Using (1), (3) and (6) we find that the relation to be proved is By substituting (13) and applying Lemma 3, this is simplified to We may drop θ(δ i ≥ 1) because if δ i = 0, the weight condition (i) α i + β i − γ i + 1 = 0 enforces 1 − q βi−γi+1 = 0. Similarly θ(γ i ≥ 1) can also be discarded. Then we are left to show This is easily checked by using the weight condition (ii).

Stochastic
where the sum γ,δ is taken over γ ∈ B l , δ ∈ B m as in (6). The last equality in (16) is derived by using α i + β i = γ i + δ i . We also introduceŠ(z) = P S(z). The both S(z) andŠ(z) will be called the stochastic R matrix or just S matrix for short.
Proposition 4. The S matrix satisfies the inversion relationŠ l,m (z)Š m,l (z −1 ) = id Vm⊗V l and the Yang- The inversion relation is obvious. Consider the Yang-Baxter equation depicted in (10). In view of the last expression in (16) we concern the sum of the three η's on each side: The both are easy to verify by using the weight conservation condition.
1 ), the following relation is valid: where F is a known function and a 2 , b 2 are determined from with the power ω given by The most notable feature of the S matrix is the following.
Theorem 6. For any l, m ∈ Z ≥1 , the S matrix S(z) = S l,m (z) enjoys the sum-to-unity property: Note that there is no constraint l ≤ m for this assertion.
where λ, µ are generic parameters. By the definition Φ q (γ|β; λ, µ) = 0 unless γ ≤ β. Note that β and γ here are n-component arrays rather than n + 1 as opposed to the indices in S(z) α,β γ,δ . In the case n = 1, the power ξ vanishes and the function (19) reproduces [27, eq.(8)] as which is known as the weight function associated with q-Hahn polynomials. As it turns out, our U q (A n ) generalization (19) arises as the special value of the S matrix.
In view of Proposition 7, Theorem 6 is rephrased in terms of an n-component array β as the identity for any positive integers l, m such that l ≤ m. One may remove the constraint |γ| ≤ l in the sum since the summand vanishes otherwise.

2.3.
Regarding λ = q −l , µ = q −m as parameters. Proposition 4, Theorem 6 and Proposition 7 remain valid even when we replace q −l and q −m with parameters λ and µ as we shall explain below. In this subsection, we fix q, z, set λ = q −l , µ = q −m and regard λ, µ as variables. Note that the action of n ) on V l ⊗ V m gives rise to Laurent polynomials in λ, µ. We wish to show that the matrix elements R(z) γ,δ α,β are rational functions in λ, µ. Since l varies, we utilize α = (α 1 , . . . , α n ) ∈ Z n ≥0 as a labeling of basis vectors |α 1 , . . . , α n of V l . So is β for V m . Thus the symbol |0 which is the abbreviation of |0, . . . , 0 is to be understood as an appropriate highest weight vector appearing in (5). Due to the weight conservation property R(z) γ,δ α,β = 0 unless α + β = γ + δ, we concentrate on the case when α + β = γ + δ = ̟ for some fixed weight ̟ ∈ Z n ≥0 . Take N such that |̟| < N and then take l, m such that N < l, m.
Once we understand that R(z) γ,δ α,β is a rational function in λ = q −l , µ = q −m , we can show that the Yang-Baxter equation (8) or (9) is satisfied as an identity of matrix-valued rational functions in κ = q −k , λ = q −l , µ = q −m . To see this, fix a weight ̟ = α + β + γ and take an integer N such that |̟| < N . Consider a particular coefficient of both sides of (9) applied to a vector |α ⊗ |β ⊗ |γ such that α + β + γ = ̟. Eliminating the denominators, both sides are polynomials in κ, λ, µ. We know that substituting κ = q −k , λ = q −l , µ = q −m where k, l, m are integers such that N < k, l, m, both sides are equal to each other. Since we can choose infinitely many independent integers for k, l, m, this identity must be the one as polynomials in κ, λ, µ.
It satisfies the Yang-Baxter equation, the inversion relation and the sum-to-unity condition: They are consequences of Proposition 4, Theorem 6 and the argument in Section 2.3.

Remark 8.
As seen from (19) and (26), the specialized S matrix S(λ, µ) is a solution of the Yang-Baxter equation without "difference property", meaning that its dependence on λ and µ is not only through the combination λ/µ.
We set ν = µ/λ. The case n = 1 is equivalent to k j=0 ν k−j (ν; q) j k j q = 1 for ∀k ∈ Z ≥0 , which is easily verified. We invoke the induction on n. Defineβ = (β 2 , . . . , β n ) and similarlyγ. From (20) one has where the first and the second equalities are due to the induction assumption at n = n − 1 and n = 1, respectively.

Stochastic models
In this and the next section, we will be exclusively concerned with systems with the periodic boundary condition.
3.1. Commuting transfer matrices. We construct two types of commuting transfer matrices based on the stochastic R matrices S(z) and S(λ, µ). To extract Markov processes from them one has to find an appropriate specialization that fulfills the basic axioms of the Markov matrix. This issue will be argued in Section 3.2, 3.3 and 3.4.
In this way we obtain a commuting family of evolution systems associated with (39) among which the cases l ≤ min{m 1 , . . . , m L } can be regarded as discrete time Markov processes.
The diagram (35) is naturally interpreted in terms of n species of particles obeying stochastic dynamics on the one-dimensional lattice. It is supplemented with an extra lane (auxiliary space) which particles get on or get off when they leave or arrive at a site. The local situation at the i-th site from the left with β i = (β i,1 , . . . , β i,n+1 ) ∈ B mi and γ i = (γ i,1 , . . . , γ i,n+1 ) ∈ B l is depicted as follows.
γi−1,n n · · · · n ❄ ✲ γi,n n · · · n γi−1,1 1 · · · · 1 ❄ ✲ γi,1 1 · · · 1 βi,1 1 · · · · ·1 βi,2 2 · · · · ·2 · · · βi,n n · · · · ·n βi,n+1 The site i can accommodate up to m i particles. The β i,a is the number of particles of species a for a ∈ [1, n] and the vacancy for a = n + 1. Among the β i,a particles of species a, γ i,a (≤ β i,a ) of them are moving out to the right while γ i−1,a are moving in from the left. The former event contributes the factor Φ q 2 (γ i |β i ; q −2l , q −2mi ) to the total rate. The number of particles on the extra lane is at most l at every border of the adjacent sites. Such a dynamics is closely parallel with its deterministic counterpart, an integrable cellular automaton known as box-ball system with capacity-l carrier and capacity-m i box at site i. See [13] and references therein.

3.3.
Discrete time Markov chain without particle number constraint. Let us proceed to the system associated with the transfer matrix (37) whose evolution is governed by Although this is an equation in an infinite-dimensional vector space, it actually splits into finite-dimensional subspaces specified by the particle content as T(λ|µ 1 , . . . , µ L ) preserves the weight. One can satisfy the axioms (i) and (ii) for the discrete time Markov process stated after (38). In fact, the non-negativity (i) holds if Φ q (γ|β; λ, µ i ) ≥ 0 for all i ∈ [1, L]. This is achieved by taking 0 < µ ǫ i < λ ǫ < 1, q ǫ < 1 in the either alternative ǫ = ±1. The sum-to-unity condition (ii) α1,...,αL T α1,...,αL β1,...,βL = 1 is valid thanks to (31). The resulting stochastic dynamical system is parallel with the previous one associated with T (l|m 1 , . . . , m L ) under the formal correspondence λ = q −l , µ i = q −mi . See (27). The most notable difference, however, is that for the generic λ, µ 1 , . . . , µ L in the present setting, there is no upper bound on the number of particles occupying a site i nor those hopping from i to i + 1 (i mod L). It is described by the n-component arrays β i , γ i ∈ Z n ≥0 with the local transition rate factor Φ q (γ i |β i ; λ, µ i ) (19). When n = 1 and µ 1 = · · · = µ L , such a system was introduced originally in [27]. As discussed therein, one can control the number of hopping particles in various ways by specializing λ, µ.
It turns out that the two models can be identified through a certain transformation. To explain it, let us exhibit the regime/parameter dependence as H(ǫ, q, µ) andĤ(ǫ, q, µ). The key to the equivalence is the identity which can be directly checked from (43) and (53). Comparing (50) and (57) by applying (61), one finds that the two Markov matrices are linked as Here P = P −1 ∈ End(W ⊗L ) is the "parity" operator reversing the sites as P|σ 1 , . . . , σ L = |σ L , . . . , σ 1 which adjusts the directions of γ-arrows in (50) and (57). Thus studying either one of H orĤ for the two regimes ǫ = ±1 is equivalent to treating the two models concentrating on either one of the regimes. It is intriguing that two members in the commuting family {τ (λ|µ)} with respect to λ are linked by the relation like (62). We will explain the coincidence of the spectra implied by it also at the level of Bethe ansatz around (76).
Let us include a comment on the model corresponding toĤ(1, q, 0). From (59), the relevant transition rate is It defines a one-parameter family of integrable n-species TAZRP for 0 ≤ q < 1. In particular at q = 0, the local dynamics is frozen to the situation 1≤i<j≤n γ i (α j − γ j ) = 0. To digest this constraint, let s be the minimum of the species of the particles that are jumping out. Namely, s ∈ [1, n] is the smallest among those satisfying γ s > 0. Then the above condition implies γ a = α a for all a ∈ [s + 1, n]. It means that all the particles with species larger than s must also be jumping out simultaneously. In other words, larger species particles always have the priority in the multiple particle jumps, and all such events have an equal rate. Such a stochastic dynamics exactly coincides with the n-species TAZRP in [17] with the homogeneous choice of the parameters w 1 = · · · = w n therein. Thus (63) can be viewed as defining an integrable q-melting of it. Remark 10. Our particle interpretation here and the previous subsection is entirely based on regarding the first n components in the arrays α = (α 1 , . . . , α n+1 ) as the number of n species of particles. However it is a matter of option which components one regards so. Changing them would lead to apparently different variety of stochastic dynamics of multispecies particle systems.
Examples of actual eigenvalues and Bethe roots are available in Example 12.
In general let us separate the sum (67) into two cases according to a l = n + 1 or a l ≤ n. The former consists of the single term corresponding to a 1 = · · · = a l = n + 1, whereas the latter always contains where X(z) is a rational function without a pole at z = q l in general.

4.2.
Spectrum of T (l|m 1 , . . . , m L ). Now we are ready to derive the spectrum of the discrete time Markov matrix T (l|m 1 , . . . , m L ) in (39). Under the specialization z = q l and w i = q mi , the second term in (69) vanishes, therefore the eigenvalue formula takes the factorized form Λ(l, q l | m1,...,mL q m 1 ,...,q m L ) = in terms of u (n) j 's that are determined from the specialized Bethe equation: .
(71) 4.3. Spectrum of T(λ|µ 1 , . . . , µ L ). The Markov transfer matrix T(λ|µ 1 , . . . , µ L ) was defined in (37). Below we write down a natural extrapolation of the results in the previous subsection in view of the correspondence (27) although their rigorous derivation is yet to be supplied. The eigenvalues Λ(λ|µ 1 , . . . , µ L ) of T(λ|µ 1 , . . . , µ L ) and the Bethe equation are given by where q 1/2 has been avoided by replacing u When n = 1, these results reduce to [27, eq.(38) and Bethe eq. on p17] by replacing (u (1) j , µ, λ) with (νu j , ν, ν/µ). From (60), eigenvalues of the continuous time Markov matrices H andĤ (denoted by the same symbols) are obtained by differentiation with respect to λ. Since the Bethe roots are independent of λ, they are given by in terms of solutions to the same Bethe equation (75). One can detect the "spectral equivalence" implied by (62) also from the Bethe ansatz result here. Denote the system of Bethe equations (75) symbolically by B({u (a) j }, q, µ) and the eigenvalue formulas (76) by H({u For n = 1, there remains no other Bethe equation to be solved, indicating that the steady state is uniform (or possesses a product measure at most) under the periodic boundary condition as emphasized in [27,10]. In general the steady state for n ≥ 2 is nontrivial. However at least on the level of Bethe roots, they exhibit the same simplifying feature as the n = 1 case. The following example is an exposition of this fact.
On the other hand, steady states themselves are nontrivial for multispecies case n ≥ 2.
As these examples indicate, steady states for multispecies case n ≥ 2 are involved but algebraic 10 in that no transcendental input from nontrivial solutions to the Bethe equation is required. The steady states are known to exhibit rich combinatorial and algebraic structures related to the crystal base of quantum groups and the tetrahedron equation already at q = 0 [18]. Their systematic investigation will be presented elsewhere.

Summary
In this paper we have explored new prospects of the U q (A (1) n ) quantum R matrix for the symmetric tensor representation V l ⊗ V m which have applications to integrable stochastic models in non-equilibrium statistical mechanics.
The R matrix R(z) has been shown to factorize at z = q l−m for l ≤ m from which the non-negativity is manifest in an appropriate range of the remaining parameters (Theorem 2). We have found a suitable gauge S(z) (15) of R(z) which satisfies the sum rule (Theorem 6) as well as the Yang-Baxter equation (Proposition 4). We have also introduced the specialized S matrix S(λ, µ) corresponding to the extrapolation of S(z = q l−m ) to generic l, m. It also satisfies the non-negativity, the sum rule (30), (31) and the Yang-Baxter equation without "difference property" (Remark 8).
Based on the stochastic R matrices S(z) and S(λ, µ), we have constructed new integrable Markov chains described in terms of n species of particles obeying asymmetric dynamics. They are discrete time systems with (Section 3.2) and without (Section 3.3) constraints on the number of particles at lattice sites and those hopping to the neighboring site at one time step. The other ones (Section 3.4) are n-species TAZRPs corresponding to continuous time limits of that in Section 3.3. Two such TAZRPs associated to the "Hamiltonian points" λ = 1 and λ = µ of the Markov transfer matrix are obtained and their interrelation (62) has been clarified. They admit a superposition yielding an integrable asymmetric zero range process in which n species of particles can hop to either direction (Remark 9).
The Markov matrices in these models are specializations of the commuting transfer matrices whose spectra are well-known by the Bethe ansatz in the theory of quantum integrable systems. However, the precise adjustment to the present stochastic setting demands some work. We have given the resulting Bethe eigenvalue formulas for all the models under the periodic boundary condition (Section 4). In particular, the eigenvalues relevant to the steady states are found to correspond to the trivial choice ∀Q a (z) = 1 of the Baxter Q functions. This explains the algebraic (non-transcendental) nature of the steady states from the Bethe ansatz point of view, indicating a possible alternative approach by the method of matrix products. These issues will be addressed elsewhere.
Similarly the U q (A (1)