Integrable Structure of Multispecies Zero Range Process

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic R matrices of quantum affine algebra Uq(An⁽¹⁾), matrix product construction of stationary states for periodic systems, q-boson representation of Zamolo...

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Дата:	2017
Автори:	Kuniba, A., Okado, M., Watanabe, S.
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Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Integrable Structure of Multispecies Zero Range Process / A. Kuniba, M. Okado, S. Watanabe // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ.

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spelling	irk-123456789-1486472019-02-19T01:24:46Z Integrable Structure of Multispecies Zero Range Process Kuniba, A. Okado, M. Watanabe, S. We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic R matrices of quantum affine algebra Uq(An⁽¹⁾), matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter. 2017 Article Integrable Structure of Multispecies Zero Range Process / A. Kuniba, M. Okado, S. Watanabe // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R50; 60C9 DOI:10.3842/SIGMA.2017.044 http://dspace.nbuv.gov.ua/handle/123456789/148647 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic R matrices of quantum affine algebra Uq(An⁽¹⁾), matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.
format	Article
author	Kuniba, A. Okado, M. Watanabe, S.
spellingShingle	Kuniba, A. Okado, M. Watanabe, S. Integrable Structure of Multispecies Zero Range Process Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kuniba, A. Okado, M. Watanabe, S.
author_sort	Kuniba, A.
title	Integrable Structure of Multispecies Zero Range Process
title_short	Integrable Structure of Multispecies Zero Range Process
title_full	Integrable Structure of Multispecies Zero Range Process
title_fullStr	Integrable Structure of Multispecies Zero Range Process
title_full_unstemmed	Integrable Structure of Multispecies Zero Range Process
title_sort	integrable structure of multispecies zero range process
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148647
citation_txt	Integrable Structure of Multispecies Zero Range Process / A. Kuniba, M. Okado, S. Watanabe // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kunibaa integrablestructureofmultispecieszerorangeprocess AT okadom integrablestructureofmultispecieszerorangeprocess AT watanabes integrablestructureofmultispecieszerorangeprocess
first_indexed	2025-07-12T19:52:35Z
last_indexed	2025-07-12T19:52:35Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 044, 29 pages Integrable Structure of Multispecies Zero Range Process Atsuo KUNIBA †, Masato OKADO ‡ and Satoshi WATANABE † † Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan E-mail: atsuo.s.kuniba@gmail.com, watanabe@gokutan.c.u-tokyo.ac.jp ‡ Department of Mathematics, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan E-mail: okado@sci.osaka-cu.ac.jp Received January 26, 2017, in final form June 07, 2017; Published online June 17, 2017 https://doi.org/10.3842/SIGMA.2017.044 Abstract. We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic R matrices of quan- tum affine algebra Uq ( A (1) n ) , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov–Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter. Key words: integrable zero range process; stochastic R matrix; matrix product formula 2010 Mathematics Subject Classification: 81R50; 60C99 1 Introduction Zero range processes (ZRPs) [44] model a variety of stochastic dynamics in biology, chemistry, networks, physics, traffic flows and so forth. Their rich behaviors like condensation, current fluctuations and hydrodynamic limit have been important issues in non-equilibrium physics. See for example [16, 21, 26] and references therein. This paper is a brief summary of the integrable multispecies ZRP in one dimension intro- duced and studied in the recent works [29, 34, 35]. We formulate the ZRPs via commuting Markov transfer matrices and present a matrix product formula for stationary probabilities in the periodic boundary condition. The key ingredients in these results are the stochastic R matrix and the Zamolodchikov–Faddeev (ZF) algebra. The subject lies in the intersection of quantum integrable systems and non-equilibrium statistical mechanics. As the title of the paper suggests, we will mainly focus on the former aspect, although we believe the results are essential for analyzing the physics of the model as far as the stationary properties are concerned. Quantum R matrices are solutions of the Yang–Baxter equation (YBE) [3] and play a most fundamental role in quantum integrable systems [24]. They can be systematically produced from the representation theory of quantum groups. It remains, however, a nontrivial problem if an R matrix can be made stochastic, namely whether it can be modified so as to match the basic criteria of Markov matrices which are non-negativity and total probability conservation. Our stochastic R matrices [29] fulfill the criteria. They originate in the quantum R matrix of the Drinfeld–Jimbo quantum affine algebra Uq ( A (1) n ) in the symmetric tensor representation This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html mailto:atsuo.s.kuniba@gmail.com mailto:watanabe@gokutan.c.u-tokyo.ac.jp mailto:okado@sci.osaka-cu.ac.jp https://doi.org/10.3842/SIGMA.2017.044 http://www.emis.de/journals/SIGMA/RAQIS2016.html 2 A. Kuniba, M. Okado and S. Watanabe of general degree. Plainly, they are of type A with arbitrary rank and spin, covering many examples that had been known earlier. Being higher in rank and being analytically continued in spin, it leads to systems with many kinds of particles allowing arbitrarily multiple occupancy at each lattice site, which are characteristic to multispecies ZRPs. These features are reviewed in Sections 2 and 3 based on [29]. Sections 2.3 and 3.2 also include ZRPs with a new mixed boundary condition. In Sections 4 and 5 we turn to the stationary probabilities P(σ1, . . . , σL) of a given configu- ration (σ1, . . . , σL) ∈ (Zn≥0)L in the ZRPs with the periodic boundary condition. Here n and L are the numbers of the species of particles and lattice sites, respectively. We seek the matrix product formula P(σ1, . . . , σL) = Tr(Xσ1(µ1) · · ·XσL(µL)) in terms of a collection of operators X(µ) = (Xα(µ))α∈Zn≥0 that satisfy the ZF algebra X(µ)⊗X(λ) = Š(λ, µ) [ X(λ)⊗X(µ) ] . It contains the stochastic R matrix Š(λ, µ) as the structure function. Here λ, µ can be understood as generic parameters as long as algebraic aspects are concerned, but they are restricted to real numbers in a certain range in the application to the ZRP. The ZF algebra, originally introduced in the factorized scattering theories in (1+1) dimension [18, 51], has penetrated into the matrix product method in integrable Markov processes in various guises since the 90’s. See general remarks in Section 5.1 and also [1, 6, 10, 14, 41]. We will review a q-boson representation of the ZF algebra obtained in [34, 35]. The simplest nontrivial case is n = 2 for which it is Xα1,α2(µ) = µ−α1−α2(µ; q)α1+α2 (q; q)α1(q; q)α2 (b; q)∞ (µ−1b; q)∞ kα2cα1 , where b, c, k are q-boson creation, annihilation and number operators on the Fock space and (z; q)m = m−1∏ i=0 (1 − zqi) is the q-shifted factorial. For general n, the matrix product operator Xα(µ) acts on the tensor product of 1 2n(n−1) Fock spaces. There are numerous matrix product formulas in terms of bosons known in the literature, most typically for the exclusion processes. See [1, 6, 14, 15, 30, 40] for example and references therein. Our result (Theorem 5.2) is the first example distinct from them involving a quantum dilogarithm type infinite product of q-bosons. One of the key facts in our approach is the explicit factorized formula (2.8) of an R matrix of Uq ( A (1) n ) at a special point of the spectral parameter. In the last Section 6 we seek a similar result for a family of R matrices associated with the generalized quantum groups labeled by (ε1, . . . , εn+1) ∈ {0, 1}n+1. They are constructed from (n + 1)-fold product of the solutions to the tetrahedron equation [50], a three-dimensional (3D) generalization of the YBE, called 3D R (εi = 0) and 3D L (εi = 1) [36]. The stochastic R matrix in Sections 2–5 originates in the special case ε1 = · · · = εn+1 = 0. We present the Serre type relations of the relevant generalized quantum groups explicitly and prove a similar factorized formula for (ε1, . . . , εn+1) of the form (1κ, 0n+1−κ) (0 ≤ κ ≤ n+ 1). These results are new. Their application is yet to be explored. The layout of the paper is as follows. In Section 2 we recall two kinds of stochastic R ma- trices S(z) and S(λ, µ), and construct several kinds of commuting transfer matrices from them. In Section 3 we specialize these transfer matrices to formulate integrable multispecies ZRPs. They include discrete and continuous time models with both periodic and mixed boundary con- ditions. The latter is new. In Section 4 stationary states of these ZRPs are studied, and its matrix product construction is linked to the ZF algebra for the models with periodic boundary Integrable Structure of Multispecies Zero Range Process 3 condition. In Section 5 we make general remarks on ZF algebra and give a q-boson represen- tation when the structure function is the stochastic R matrix S(λ, µ). It yields the stationary probabilities in the matrix product form for the associated n-species ZRP. This part is a re- view of [34, 35]. In Section 6 we extend the factorization (2.8) to the R matrices for a class of generalized quantum groups. The result is presented with some background connected to the tetrahedron equation [36]. Section 7 is a short summary. Appendix A contains the explicit form of the quantum R matrix for the generalized quantum group UA(1, 1, 0). Throughout the paper we use the notation Zn = Z/nZ, θ(true) = 1, θ(false) = 0, the q-shifted factorial (z)m = (z; q)m = m−1∏ j=0 (1 − zqj) and the q-binomial ( m k ) q = θ(k ∈ [0,m]) (q)m (q)k(q)m−k . The symbols (z)m appearing in this paper always mean (z; q)m. For integer arrays α = (α1, . . . , αm), β = (β1, . . . , βm) of any length m, we write \|α\| = α1 + · · ·+αm and the Kronecker delta as δα,β = δαβ = m∏ i=1 θ(αi = βi). The letter δ will also be used extensively to mean a local state and in such circumstances we will use the notation θ(α = β) more frequently than δα,β to avoid confusion. The relation α ≤ β or equivalently β ≥ α is defined by β − α ∈ Zm≥0. We often denote by 0 or 0m to mean (0, . . . , 0) ∈ Zm≥0. While preparing the text, we were informed of the paper [27], where the author obtains Markov duality functions for the models treated in this paper. 2 Commuting transfer matrices 2.1 Stochastic R matrices Let us recall the stochastic R matrices S(z) and S(λ, µ) [29] associated with the Drinfeld–Jimbo quantum affine algebra Uq ( A (1) n ) . They are constructed by suitably modifying the quantum R matrix characterized by (6.12). For l ∈ Z≥1, introduce the vector space Vl whose basis is labeled with the set Bl as Bl = { α = (α1, . . . , αn+1) ∈ Zn+1 ≥0 \| \|α\| = l } , Vl = ⊕ (α1,...,αn+1)∈Bl C\|α1, . . . , αn+1〉. (2.1) We write \|α1, . . . , αn+1〉 simply as \|α〉. There is an algebra homomorphism Uq ( A (1) n ) → End(Vl) called the symmetric tensor representation of degree l depending on a spectral parameter. We are concerned with the standard quantum R matrix R(z) = Rl,m(z) living in End(Vl ⊗ Vm). Leaving the representation theoretical background to Section 6, we present an explicit formula: R(z)(\|α〉 ⊗ \|β〉) = ∑ γ∈Bl,δ∈Bm R(z)γ,δα,β\|γ〉 ⊗ \|δ〉, (2.2) R(z)γ,δα,β = z−m(ql−mz; q2)m+1 (ql−m+2z−1; q2)m ∑ c0,...,cn∈Z≥0 zc0Rγ1,δ1,c0α1,β1,c1 · · ·Rγn,δn,cn−1 αn,βn,cn R γn+1,δn+1,cn αn+1,βn+1,c0 , (2.3) R a,b,c i,j,k = δa+bi+j δ b+c j+kq ik+b ∮ du 2πiub+1 (−q2+a+cu; q2)∞(−q−i−ku; q2)∞ (−qa−cu; q2)∞(−qc−au; q2)∞ ∈ Z[q]. (2.4) The integral encircles the origin u = 0 anti-clockwise to pick the residues. z is called the spectral parameter. Explicit formulas for Ra,b,ci,j,k are available for example in [36, equation (2.2)]. The fact that R a,b,c i,j,k ∈ Z[q] can be seen from them. Owing to the factor δa+bi+j δ b+c j+k in (2.4), the sum (2.3) consists of finitely many terms and R(z)γ,δα,β is a rational function of z and q. The prefactor in (2.3) has been chosen so as to achieve the normalization (6.15), which will ultimately lead to (2.7) related to the total probability conservation. 4 A. Kuniba, M. Okado and S. Watanabe The set of q-polynomials {Ra,b,ci,j,k} form a solution of the tetrahedron equation [50] having an origin in the quantized coordinate ring of SL3 [25]. It was stated that the composition (2.3) yields the quantum R matrix in [4] and proved in [33, Appendix B]. The formula (2.4) is due to [43]. See Section 6 and [33, 36] for a further explanation and generalization. For recent progress on evaluating the sum (2.3), we refer to [9]. The first stochastic R matrix S(z) = Sl,m(z) ∈ End(Vl ⊗ Vm) is obtained just by taking the stochastic gauge of R(z) as follows: S(z)(\|α〉 ⊗ \|β〉) = ∑ γ∈Bl,δ∈Bm S(z)γ,δα,β\|γ〉 ⊗ \|δ〉, S(z)γ,δα,β = qηR(z)γ,δα,β, η = ∑ 1≤i<j≤n+1 (δiγj − αiβj). (2.5) Example 2.1. Consider the simplest example S(z) = S1,1(z). We denote S(z) ei,ej ek,el simply by S(z)i,jk,l, where ei is the i th basis vector defined in (6.13). By the graphical representa- tion (2.14), nonzero elements are given by where 1 ≤ i 6= j ≤ n+1. They lead to the n-species symmetric simple exclusion process at q = 1 and asymmetric simple exclusion process for q 6= 1. See Section 3.3. Theorem 2.2 ([29]). Set zi,j = zi/zj. Then the following equalities are valid: YBE : Sk,l1,2(z1,2)S k,m 1,3 (z1,3)S l,m 2,3 (z2,3) = Sl,m2,3 (z2,3)S k,m 1,3 (z1,3)S k,l 1,2(z1,2), (2.6) sum-to-unity : ∑ γ∈Bl, δ∈Bm Sl,m(z)γ,δα,β = 1, ∀ (α, β) ∈ Bl ×Bm, (2.7) factorization : Sl,m ( z = ql−m )γ,δ α,β = δγ+δα+βΦq2 ( γ̄\|β̄; q−2l, q−2m ) , l ≤ m, (2.8) where γ̄ = (γ1, . . . , γn), β̄ = (β1, . . . , βn) ∈ Zn≥0 are the arrays γ, β with the (n+1)-th component dropped. Here Sk,l1,2(z1,2) for instance denotes the matrix that acts as Sk,l(z1,2) on the first and the second components from the left in Vk ⊗ Vl ⊗ Vm. The Sl,m(z)γ,δα,β is an element of the mat- rix Sl,m(z). In (2.6) and (2.7), there is no constraint like l ≤ m in (2.8). The Φq(γ\|β;λ, µ) appearing in (2.8) is the function of n-component arrays β, γ and parameters q, λ, µ defined by Φq(γ\|β;λ, µ) = qϕ(β−γ,γ) (µ λ )\|γ\| (λ; q)\|γ\|( µ λ ; q)\|β\|−\|γ\| (µ; q)\|β\| n∏ i=1 ( βi γi ) q , ϕ(α, β) = ∑ 1≤i<j≤n αiβj . (2.9) By the definition Φq(γ\|β;λ, µ) = 0 unless γ ≤ β. The modification by the factor qη in (2.5) does not spoil the YBE. The point is that it can be so chosen that the sum-to-unity property (2.7) Integrable Structure of Multispecies Zero Range Process 5 holds. It will eventually lead to the total probability conservation in the relevant stochastic models in what follows. The factorization (2.8) at the special point z = ql−m is also nontrivial, and assures the non-negativity of the transition rate manifestly in an appropriate range of parameters. We will generalize a formula like (2.8) to a wider class of R matrices in Section 6. The second stochastic R matrix S(λ, µ) is extracted essentially from (2.8)\|q2→q by regar- ding q−l, q−m as parameters λ, µ. It is a linear operator on W ⊗ W with W defined by W = ⊕ (α1,...,αn)∈Zn≥0 C\|α1, . . . , αn〉. The basis \|α1, . . . , αn〉 here is labeled with an n-component array as opposed to (2.1), but we also denote it by the same symbol \|α〉 for simplicity. Then S(λ, µ) is defined by S(λ, µ)(\|α〉 ⊗ \|β〉) = ∑ γ,δ∈Zn≥0 S(λ, µ)γ,δα,β\|γ〉 ⊗ \|δ〉, (2.10) S(λ, µ)γ,δα,β = δγ+δα+βΦq(γ\|β;λ, µ), (2.11) where Φq(γ\|β;λ, µ) is given by (2.9). We refer to the property S(λ, µ)γ,δα,β = 0 unless α+β = γ+δ as weight conservation. The sum (2.10) is finite due to the weight conservation. In fact, the direct sum decomposition W ⊗W = ⊕ κ∈Zn≥0 (⊕ α+β=κC\|α〉⊗ \|β〉 ) holds and S(λ, µ) splits into the corresponding submatrices. Note that the “difference property” commonly known for the original quantum R matrix and also S(z) in (2.5) has been lost and S(λ, µ) = S(cλ, cµ) does not hold. Theorem 2.3 ([29]). The following equalities hold: YBE : S1,2(ν1, ν2)S1,3(ν1, ν3)S2,3(ν2, ν3) = S2,3(ν2, ν3)S1,3(ν1, ν3)S1,2(ν1, ν2), (2.12) sum-to-unity : ∑ γ,δ∈Zn≥0 S(λ, µ)γ,δα,β = 1, ∀α, β ∈ Zn≥0. (2.13) In particular (2.13) implies∑ γ∈Zn≥0 Φq(γ\|β;λ, µ) = 1, ∀β ∈ Zn≥0, where the summands are nonzero only for γ ≤ β. Both matrices S(z) and S(λ, µ) also satisfy the inversion relation. We will depict either matrix elements S(z)γ,δα,β or S(λ, µ)γ,δα,β by the vertex diagram (2.14) We note basic properties: Sm,m(1)(\|α〉 ⊗ \|β〉) = \|β〉 ⊗ \|α〉, S(λ, λ)(\|α〉 ⊗ \|β〉) = \|β〉 ⊗ \|α〉. (2.15) The former originates in (6.12) with (l, x) = (m, y) and the latter can be checked easily from (2.11) and (2.9). 2.2 Commuting transfer matrices with periodic boundary condition For l,m1, . . . ,mL ∈ Z≥1 and parameters z, w1, . . . , wL, set T ( l, z\|m1,...,mL w1,...,wL ) = TrVl ( Sl,mL0,L (z/wL) · · ·Sl,m1 0,1 (z/w1) ) ∈ End ( Vm1 ⊗ · · · ⊗ VmL ) . (2.16) 6 A. Kuniba, M. Okado and S. Watanabe In the terminology of the quantum inverse scattering method, it is the row transfer matrix of the Uq ( A (1) n ) vertex model of size L with periodic boundary condition whose quantum space is Vm1 ⊗ · · · ⊗ VmL with inhomogeneity w1, . . . , wL and the auxiliary space Vl signified by 0 with spectral parameter z. The Sl,mi0,i (z/wi) is the matrix (2.5) acting as Sl,mi(z/wi) on Vl⊗Vmi and as identity elsewhere. The dependence on q has been suppressed in the notation. It has the difference property T ( l, z\|m1,...,mL w1,...,wL ) = T ( l, az\| m1,...,mL aw1,...,awL ) . Thanks to the YBE (2.6), it forms a commuting family [3]:[ T ( l, z\|m1,...,mL w1,...,wL ) , T ( l′, z′\|m1,...,mL w1,...,wL )] = 0. The T = T (l, z\|m1,...,mL w1,...,wL ) acts on a base vector representing a row configuration as1 T \|β1, . . . , βL〉 = ∑ αi∈Bmi Tα1,...,αL β1,...,βL \|α1, . . . , αL〉 ∈ Vm1 ⊗ · · · ⊗ VmL . (2.17) The matrix element is given by Tα1,...,αL β1,...,βL = ∑ γ0∈Bl M ( γ0\|α1,...,αL β1,...,βL \|γ0 ) , where the summands are defined by concatenation of the diagram (2.14) as follows: (2.18) We have suppressed the dependence on (l, z) and (mi, wi) attached to the horizontal and vertical arrows. By the construction T satisfies the weight conservation: Tα1,...,αL β1,...,βL = 0 unless α1 + · · ·+ αL = β1 + · · ·+ βL ∈ Zn+1 ≥0 . (2.19) Next we proceed to the transfer matrix associated with S(λ, µ) in (2.10): T(λ\|µ1, . . . , µL) = TrW (S0,L(λ, µL) · · · S0,1(λ, µ1)) ∈ End ( W⊗L ) , (2.20) where the notations are similar to (2.16). Let Tα1,...,αL β1,...,βL be its matrix element defined analogously to (2.17). It is specified as T α1,...,αL β1,...,βL = ∑ γ0∈Zn≥0 M ( γ0\|α1,...,αL β1,...,βL \|γ0 ) , where M ( γ0\|α1,...,αL β1,...,βL \|γL ) is defined by (2.18) if the i-th vertex from the left is regarded as S(λ, µi) γi,αi γi−1,βi in (2.11) and αi’s and the sum over γ1, . . . , γL−1 are taken from Zn≥0. The horizontal arrow should then be understood as being associated with λ as well although it is suppressed in the notation. Since the summand vanishes unless γi ≤ βi for all i, the sum (2.18) for γi ∈ Zn≥0 is finite and T(λ\|µ1, . . . , µL) is well-defined. We have the commutativity[ T(λ\|µ1, . . . , µL),T(λ′\|µ1, . . . , µL) ] = 0 (2.21) and the weight conservation analogous to (2.19). The Bethe ansatz eigenvalues of T ( l, z\|m1,...,mL w1,...,wL ) and T(λ\|µ1, . . . , µL) have been obtained in [29, Section 4]. 2.3 Commuting transfer matrices with mixed boundary condition Let us present a simple example of commuting transfer matrices having mixed boundary condi- tions. Let 〈γ\| with γ ∈ Bl be the basis of the dual of Vl such that 〈γ\|γ′〉 = δγ,γ′ . Define T̃ ( i, l, z\|m1,...,mL w1,...,wL ) = ∑ γ∈Bl 〈γ\|Sl,mL0,L (z/wL) · · ·Sl,m1 0,1 (z/w1)\|lei〉 ∈ End (Vm1 ⊗ · · · ⊗ VmL) , 1We warn that \|α1, . . . , αL〉 with αi = (αi,1, . . . , αi,n+1) ∈ Bmi here is different from the one in (2.1). Integrable Structure of Multispecies Zero Range Process 7 where 〈γ\| and \|lei〉 are regarded as sitting at the 0-th tensor component. The vector \|lei〉 is defined by (6.13). The matrix element is given by T̃α1,...,αL β1,...,βL = ∑ γ∈Bl M ( lei\|α1,...,αL β1,...,βL \|γ ) . In the diagram (2.18) it corresponds to the fixed boundary condition γ0 = lei on the left and the free boundary condition γL = γ on the right. Schematically we have where • is attached to emphasize the sum2. It forms a commuting family:[ T̃ ( i, l, z\|m1,...,mL w1,...,wL ) , T̃ ( i, l′, z′\|m1,...,mL w1,...,wL )] = 0. This is most easily seen graphically as follows: The top left diagram depicts the product T̃ ( i, l, z\|m1,...,mL w1,...,wL ) T̃ ( i, l′, z′\|m1,...,mL w1,...,wL ) and so does the bottom right (with opposite order). The first equality is by the normalization Rl ′,l(z′/z)(\|l′ei〉⊗ \|lei〉) = \|l′ei〉⊗\|lei〉 (6.15), the second is by the YBE (2.6) and the last is due to the sum-to-unity property (2.7). The analogous transfer matrix can also be constructed from S(λ, µ) by modifying (2.20) as T̃(λ\|µ1, . . . , µL) = ∑ γ∈Zn≥0 〈γ\|S0,L(λ, µL) · · · S0,1(λ, µ1)\|0〉 ∈ End ( W⊗L ) . Here \|0〉 = \|0, . . . , 0〉 ∈W and 〈γ\| is the basis of the dual of W such that 〈γ\|γ′〉 = δγ,γ′ . We have the commuting family[ T̃(λ\|µ1, . . . , µL), T̃(λ′\|µ1, . . . , µL) ] = 0 by the same token owing to the normalization S(λ, µ)(\|0〉 ⊗ \|0〉) = \|0〉 ⊗ \|0〉, the YBE (2.12) and the sum-to-unity (2.13). 3 Integrable multispecies zero range process 3.1 Discrete time Markov process In the previous section we have introduced four kinds of commuting transfer matrices. Here we design their specializations to be called Markov transfer matrices that give rise to discrete time Markov processes. Denoting the time variable by t we consider the four systems endowed with the following time evolutions: \|P (t+ 1)〉 = T ( l, ql\| m1,...,mL qm1 ,...,qmL ) \|P (t)〉 ∈ Vm1 ⊗ · · · ⊗ VmL , (3.1) \|P (t+ 1)〉 = T̃ ( i, l, ql\| m1,...,mL qm1 ,...,qmL ) \|P (t)〉 ∈ Vm1 ⊗ · · · ⊗ VmL , (3.2) \|P (t+ 1)〉 = T(λ\|µ1, . . . , µL)\|P (t)〉 ∈W⊗L, (3.3) \|P (t+ 1)〉 = T̃(λ\|µ1, . . . , µL)\|P (t)〉 ∈W⊗L. (3.4) 2Of course there are also many other edges where the sum is to be taken. 8 A. Kuniba, M. Okado and S. Watanabe Let us write the action of these transfer matrices on the respective base vectors uniformly as T \|β1, . . . , βL〉 = ∑ α1,...,αL Tα1,...,αL β1,...,βL \|α1, . . . , αL〉. Then (3.1)–(3.4) can be regarded as the master equation of a Markov process with discrete time t if the following conditions are satisfied: (i) non-negativity: Tα1,...,αL β1,...,βL ∈ R≥0 for any (α1, . . . , αL) and (β1, . . . , βL), (ii) sum-to-unity: ∑ α1,...,αL Tα1,...,αL β1,...,βL = 1 for any (β1, . . . , βL). The latter represents the total probability conservation. Proposition 3.1. The conditions (i) and (ii) are satisfied if l ≤ min{m1, . . . ,mL} and q ∈ R>0 for the systems (3.1) and (3.2), and if 0 < µεi < λε < 1, 0 < qε < 1 for ε = ±1 for the systems (3.3) and (3.4). The proof was given for (3.1) and (3.3) in [29]. For (3.2) and (3.4), the proof is quite similar. In fact the non-negativity follows from the factorized explicit formulas (2.8), (2.9), (2.11), and the sum-to-unity property (ii) does from (2.7) and (2.13). Thus we have constructed, in the regimes specified in Proposition 3.1, commuting fami- lies of discrete time Markov processes labeled with l in (3.1), (3.2) (for each i) and with λ in (3.3), (3.4). For an interpretation in terms of stochastic dynamics of multispecies particles, see [29, Section 3.2]. In the systems (3.2) and (3.4), the weight, i.e., the number of particles, is not conserved. Remark 3.2. The sum-to-unity (2.7), (2.13) of the stochastic R matrices alone does not neces- sarily imply the corresponding property (ii) for the transfer matrices if one stays in the periodic boundary condition. The latter can be established by also using the independence of the matrix elements of the NW indices in (2.14), i.e., α, δ in (2.8) and (2.11) except the weight conservation factor δγ+δα+β. See the explanation after equation (39) in [29]. Thus the specialization z = ql−m in (2.8) achieves the double benefit; the factorization manifesting the non-negativity and the ‘NW-freeness’ making the sum-to-unity of R matrices propagate to the transfer matrices. Remark 3.3. In the evolution system (3.1), suppose that m1, . . . ,mL are all distinct. Set Ti = T ( mi, q mi \| m1,...,mL qm1 ,...,qmL ) = TrVmi ( Smi,mL0,L ( qmi−mL ) · · ·Smi,mi0,i (1) · · ·Smi,m1 0,1 ( qmi−m1 )) for 1 ≤ i ≤ L. By applying the left relation in (2.15) to Smi,mi0,i (1), one sees that the auxiliary space ‘merges’ into the quantum space, therefore the commuting time evolutions T1, . . . , TL can apparently be described without the former space as illustrated by the following diagrams3 for L = 3: If each arrow is formally viewed as a particle, they move around periodically to come back to the original position thereby ‘stirring’ themselves. Such a particle system appeared first in Yang’s analysis on the 1D delta-function interaction gas [49, equation (14)]. In our setting, Tp with p such that mp = min{m1, . . . ,mL} fulfills the conditions (i), (ii) in the above, which may therefore be recognized as stochastic Yang’s system. The system (3.3) also contains the similar Yang’s system at λ = µi due to the right relation in (2.15). 3In case m1, . . . ,mL are not distinct, the commutativity still holds, but in the corresponding diagram “partic- les” do not come back to the original position. Integrable Structure of Multispecies Zero Range Process 9 3.2 Continuous time Markov process In this subsection we focus on the systems (3.3) and (3.4) with the homogeneous choice of the parameters µ1 = · · · = µL = µ: T(λ\|µ) = T(λ\|µ, . . . , µ), T̃(λ\|µ) = T̃(λ\|µ, . . . , µ). From the special values Φq(γ\|β; 1, µ) = δγ,0 and Φq(γ\|β;µ, µ) = δγ,β, one can deduce that T(1\|µ) = T̃(1\|µ) = idW⊗L and T(µ\|µ) is a cyclic shift; T(µ\|µ)\|α1, . . . , αL〉 = \|αL, α1, . . . , αL−1〉. Using these facts we take the logarithmic derivatives (ε = ±1) Hr = −εµ−1∂ log T(λ\|µ) ∂λ ∣∣∣∣ λ=1 , Hl = εµ ∂ log T(λ\|µ) ∂λ ∣∣∣∣ λ=µ , (3.5) H̃ = −εµ−1∂ log T̃(λ\|µ) ∂λ ∣∣∣∣∣ λ=1 (3.6) according to Baxter’s formula (cf. [29, Section 3.4] and [3, equation (10.14.20)]), a standard pre- scription to generate spin chain Hamiltonians from vertex models in equilibrium. An intriguing aspect in our setting is the presence of two such ‘Hamiltonian points’ λ = 1 and λ = µ for the periodic case leading to the operators expressed as the sum of local terms4: Hr = ∑ i∈ZL hr,i,i+1, Hl = ∑ i∈ZL hl,i,i+1, H̃ = ∑ 1≤i≤L−1 hr,i,i+1 + h̃L. (3.7) Here hr,i,i+1, hl,i,i+1 act as hr, hl ∈ End(W ⊗W ) on the i-th and the (i+ 1)-th components and as the identity elsewhere. The h̃L acts as h̃ ∈ EndW on the L-th component and as the identity elsewhere. They are described explicitly as (0n = (0, . . . , 0) ∈ Zn≥0) hr\|α, β〉 = ε ∑ γ∈Zn≥0\{0n} qϕ(α−γ,γ)µ\|γ\|−1(q)\|γ\|−1 (µq\|α\|−\|γ\|)\|γ\| n∏ i=1 ( αi γi ) q \|α− γ, β + γ〉 − ε \|α\|−1∑ i=0 qi\|α, β〉 1− µqi , (3.8) hl\|α, β〉 = ε ∑ γ∈Zn≥0\{0n} qϕ(γ,β−γ)(q)\|γ\|−1 (µq\|β\|−\|γ\|)\|γ\| n∏ i=1 ( βi γi ) q \|α+ γ, β − γ〉 − ε \|β\|−1∑ i=0 \|α, β〉 1− µqi , (3.9) h̃\|α〉 = ε ∑ γ∈Zn≥0\{0n} qϕ(α−γ,γ)µ\|γ\|−1(q)\|γ\|−1 (µq\|α\|−\|γ\|)\|γ\| n∏ i=1 ( αi γi ) q \|α− γ〉 − ε \|α\|−1∑ i=0 qi\|α〉 1− µqi . (3.10) The matrix h̃ has the form obtainable from (3.8) by removing the dependence on β. Consider the time evolution equation d dt \|P (t)〉 = H\|P (t)〉 ∈W⊗L, H = Hr, Hl, H̃. (3.11) Denote the action of the matrices H = Hr, Hl, H̃ on the base vectors uniformly as H\|β1, . . . , βL〉 = ∑ α1,...,αL Hα1,...,αL β1,...,βL \|α1, . . . , αL〉. The equation (3.11) can be viewed as the master equation of a continuous time Markov process if the following conditions are satisfied: (i)′ non-negativity: Hα1,...,αL β1,...,βL ∈ R≥0 for any pair such that (α1, . . . , αL) 6= (β1, . . . , βL), (ii)′ sum-to-zero: ∑ α1,...,αL Hα1,...,αL β1,...,βL = 0 for any (β1, . . . , βL). 4As it turns out, the subscripts in Hr and Hl signify the right and the left hopping dynamics of particles. 10 A. Kuniba, M. Okado and S. Watanabe The latter represents the total probability conservation. From (3.8)–(3.10) we note that for a fixed initial condition there is a constant M satisfying \|Hα1,...,αL α1,...,αL \| < M uniformly for all (α1, . . . , αL). Proposition 3.4 ([29]). For all the matrices Hr, Hl and H̃, the condition (ii)′ is satisfied. So is the condition (i)′ if 0 < qε, µε < 1 for ε = ±1. Thus we have obtained, in the regimes specified in Proposition 3.4, the Markov processes corresponding to the continuous time variant of (3.3) and (3.4). The commutativity (2.21) and the construction (3.5) imply [Hr, Hl] = 0. Therefore they share the same eigenvectors with the superposition H(a, b, ε, q, µ) = aHr(ε, q, µ) + bHl(ε, q, µ) with the coefficients a, b to be taken positive in the Markov process context. A curious symmetry H(a, b,−ε, q−1, µ−1) = PH(µb, µa, ε, q, µ)P−1 is known to hold [29, Remark 9], where P = P−1 ∈ End(W⊗L) is the ‘parity’ operator reversing the sites as P\|α1, . . . , αL〉 = \|αL, . . . , α1〉. The Markov processes (3.11) with H = Hr, Hl are naturally viewed as the stochastic dy- namics of n-species of particles on a ring of length L. The base vector \|α1, . . . , αL〉 with αi = (αi,1 . . . , αi,n) ∈ Zn≥0 represents a configuration in which there are αi,a particles of species a at the i-th site. There is no constraint on the number of particles that occupy a site. The matri- ces Hr and Hl describe the stochastic hopping of them to the right and the left nearest neighbor sites, respectively. The transition rate can be read off the first terms on the r.h.s. of (3.8) and (3.9), where the array γ = (γ1, . . . , γn) specifies the numbers of particles that are jumping out. In case of the superposition H(a, b, ε, q, µ) mentioned in the above, we have a mixture of such right and left movers. Note that the rate is determined from the occupancy of the depar- ture site only and independent of that at the destination site, justifying the name “zero range process”. Here is a snapshot of the system for the n = 2 case5: The process (3.11) with H = H̃ is similarly interpreted as a stochastic particle system defined on a length L segment with boundaries. It consists of right movers only which will eventually exit from the right boundary: The integrable Markov processes constructed here and Section 3.1 cover several models stud- ied earlier. When ε = 1, µ → 0 in Hr, the nontrivial local transitions in (3.8) are limited to the case \|γ\| = 1. So if γa = 1 and the other components of γ are 0, the rate reduces to qα1+···+αa−1 1−qαa 1−q . This reproduces the n-species q-boson process in [46] whose n = 1 case further goes back to [42]. For n = 1, there are extensive list of works including [5, 7, 8, 12, 13, 22, 39, 45] for example. One can overview their interrelation in [27, Figs. 1 and 2]. When ε = 1, (µ, q) → (0, 0) in Hl, a kinematic constraint ϕ(γ, β − γ) = ∑ 1≤i<j≤n γi(βj − γj) = 0 occurs in (3.9). In 5The arrangement of particles within each site does not matter. Integrable Structure of Multispecies Zero Range Process 11 fact, in order that γa > 0 happens, the equalities γa+1 = βa+1, γa+2 = βa+2, . . . , γn = βn must hold. It means that larger species particles have the priority to jump out, which precisely re- produces the n-species totally asymmetric zero range process explored in [31, 32] after reversing the labeling of the species 1, 2, . . . , n of the particles. 3.3 Models associated with S(z) Let us remark on the models associated with the stochastic R matrix S(z) = Sl,m(z) (2.5). We refer to [27] for a further account. To the relevant transfer matrix (2.16) with the uniform choice mi = m and wi = 1, one can as- sociate the Hamiltonian H(m) = ± ∂ ∂z log T ( m, z\|m,...,m1,...,1 ) \|z=1 similarly to (3.5) and (3.7). In fact one finds H(m) = ∑ i∈ZL h(m)i,i+1 using the left property in (2.15) where the local Hamiltonian h(m)\|α, β〉 = ∑ γ,δ h(m)γ,δα,β\|δ, γ〉 is specified by h(m)γ,δα,β = ± ∂ ∂zS(z)γ,δα,β\|z=1. This is the standard derivation of an integrable ‘spin m 2 ’ Hamiltonian for Uq ( A (1) n ) by the Baxter formula (cf. [3, equation (10.14.20)]). Thanks to (2.7), H(m) enjoys the sum-to-zero property (ii)′ mentioned after (3.11). However the non-negativity (i)′ is not satisfied just by adjusting the overall sign in general for m ≥ 26. This is one of the difficulties that has been overcome by switching from Sm,m(z) to S(λ, µ). The exception is m = 1, where one sees from Example 2.1 that H(1) (with the + sign in the above) is nothing but the Markov matrix of the n-species asymmetric simple exclusion process (ASEP)7. The transition rate ri,j of \|i, j〉 → \|j, i〉 satisfies ri,j : rj,i = 1 : q2 for 1 ≤ i < j ≤ n+ 1. Many results have been obtained for the n-species ASEP with general n including, for example, matrix product stationary states [40], spectral duality with respect to the Hasse diagram of sectors [2], solutions to stochastic initial value problem on infinite lattice [47], connection to the tetrahedron equation at q = 0 [30], application to generalized Macdonald polynomials [20] and so on. 4 Stationary states 4.1 Stationary probability Here we consider the systems (3.1) and (3.3). Introduce the finite-dimensional subspaces of Vm1 ⊗ · · · ⊗ VmL and W⊗L with fixed weight: V (k) = ⊕(σ1,...,σL)∈B(k)C\|σ1, . . . , σL〉, W (k) = ⊕(σ1,...,σL)∈B(k)C\|σ1, . . . , σL〉, where the labeling sets of the bases are given by B(k) = { (σ1, . . . , σL) ∈ Bm1 × · · · ×BmL \|σ1 + · · ·+ σL = k } , (4.1) B(k) = { (σ1, . . . , σL) ∈ (Zn≥0)L \|σ1 + · · ·+ σL = k } , (4.2) where k ∈ Zn+1 ≥0 in (4.1) and k ∈ Zn≥0 in (4.2). Note that B(k) = ∅ if \|k\| > m1 + · · ·+mL. Denote the discrete time evolutions (3.1) and (3.3) simply by \|P (t + 1)〉 = T \|P (t)〉. They decompose into the equations on the subspaces V (k) for (3.1) and W (k) for (3.3), which we call sectors. If some components of k ∈ Zn≥0 are 0, the system in such a sector becomes equivalent to the one with smaller n by an appropriate relabeling of the species. In view of this, we shall concentrate with no loss of generality on the situation k ∈ Zn≥1 which we call basic sectors. 6This is also noted in [13, Remark 5.2] for m = 2. On the other hand for Sl,m(z) itself rather than its derivative, there is a range in which elements are non-negative [27, Proposition 3.6]. 7It is a model in which each site can take n+ 1 states. 12 A. Kuniba, M. Okado and S. Watanabe By definition stationary states are those invariant under T , namely \|P̄ 〉 satisfying \|P̄ 〉 = T \|P̄ 〉. There is a unique such vector \|P̄ 〉 = ∑ σ1,...,σL P(σ1, . . . , σL)\|σ1, . . . , σL〉 in each sector satisfying∑ σ1,...,σL P(σ1, . . . , σL) = 1. This is a consequence of the irreducibility of the sectors under T and the Perron–Frobenius theorem. The coefficient P(σ1, . . . , σL) is called the stationary probability. In what follows we shall abuse the terminology also to mean the unnormalized probabilities and states. We will treat the discrete time processes only since they cover the continuous time case. Thanks to the commutativity of the Markov transfer matrices, the stationary states are independent of l for (3.1) and of λ for (3.3). Before closing the subsection, we include comments on the systems (3.4) and (3.11) with H = H̃, where the number of particles is not preserved. Starting from any initial condition, any state tends to the trivial one \|0n〉 ⊗ · · · ⊗ \|0n〉, although the relaxation to it remains as an important problem. The clue to investigating it is the Bethe eigenvalues of H̃. To construct them is a feasible task as done in [29, Section 4] for the periodic case Hr and Hl. We leave it for a future study. 4.2 Some examples From now on we focus on the discrete time process defined by (3.3). The stationary state \|P̄ 〉 in the sector W (k) is characterized by \|P̄ 〉 = T(λ\|µ1, . . . , µL)\|P̄ 〉 ∈W (k). We will refer to a sector W (k) also by k = (k1, . . . , kn) ∈ Zn≥0 for simplicity. It was known in the single species case n = 1 that the stationary state possesses the product measure (cf. [17, 39] at least for the homogeneous case ∀µi = µ): P(σ1, . . . , σL) = L∏ i=1 gσi(µi), σi ∈ Z≥0, where gσi(µi) is the n = 1 case of the function gα(µ) = µ−\|α\|(µ)\|α\| n∏ i=1 (q)αi for α = (α1, . . . , αn) ∈ Zn≥0. (4.3) Such a factorization, however, is no longer valid in the multispecies case n ≥ 2 making the system nontrivial and interesting even without an introduction (cf. [15]) of a reservoir. Particles of a given species must behave under the influence of the other species acting as a nontrivial dynamical background. Example 4.1. For L = 2, n = 2 and the sector k = (1, 1), we have \|P 〉 = µ21(1− µ2)(1− qµ2)(µ1 + µ2 − 2µ2µ1)\|∅, 12〉 + µ1µ2(1− µ1)(1− µ2)(µ1 + qµ2 − µ1µ2 − qµ1µ2)\|1, 2〉+ cyclic. Here \|∅, 12〉 and \|1, 2〉 are the multiset representations of \|(0, 0), (1, 1)〉 and \|(1, 0), (0, 1)〉. For L = 3, n = 2 and the sector k = (1, 1), we have \|P 〉 = µ21µ 2 2(1− µ3)(1− qµ3)(µ1µ2 + µ1µ3 + µ2µ3 − 3µ1µ3µ2)\|∅,∅, 12) + µ21µ2µ3(1− µ2)(1− µ3)(qµ1µ2 + µ1µ3 + µ2µ3 − 2µ1µ2µ3 − qµ1µ2µ3)\|∅, 2, 1〉 + µ21µ2µ3(1− µ2)(1− µ3)(µ1µ2 + qµ1µ3 + qµ2µ3 − µ1µ2µ3 − 2qµ1µ2µ3)\|∅, 1, 2〉 + cyclic. Integrable Structure of Multispecies Zero Range Process 13 Here cyclic means the sum of terms obtained by the replacement µj → µj+i and \|σ1, . . . , σL〉 → \|σ1+i, . . . , σL+i〉 over i ∈ ZL with i 6= 0. Based on computer experiments we propose Conjecture 4.2. For the stationary state \|P̄ 〉 in any sector k, there is a normalization such that P(σ1, . . . , σL) ∈ Z≥0[q,−µ1, . . . ,−µL] for all (σ1, . . . , σL) ∈ B(k). 4.3 Matrix product construction We seek the steady state probability in the matrix product form P(σ1, . . . , σL) = Tr(Xσ1(µ1) · · ·XσL(µL)) (4.4) in terms of some operator Xα(µ) with α ∈ Zn≥0. Our strategy is to invoke the following result. Proposition 4.3. Suppose the operators Xα(µ) (α ∈ Zn≥0) obey the relation Xα(µ)Xβ(λ) = ∑ γ,δ∈Zn≥0 S(λ, µ)β,αγ,δXγ(λ)Xδ(µ). (4.5) Suppose further that dim Ker(T(µi\|µ1, . . . , µL) − 1) = 1 for some i. Then the matrix product formula (4.4) of the stationary probability holds for the system (3.3) provided that the trace is convergent and not identically zero. Proof. The stationarity T(λ\|µ1, . . . , µL)\|P̄ 〉 = \|P̄ 〉 follows from the special case λ = µi. To see this note that the latter leads to T(λ\|µ1, . . . , µL)\|P̄ 〉 = T(λ\|µ1, . . . , µL)T(µi\|µ1, . . . , µL)\|P̄ 〉 = T(µi\|µ1, . . . , µL)T(λ\|µ1, . . . , µL)\|P̄ 〉, telling that T(λ\|µ1, . . . , µL)\|P̄ 〉 ∈ Ker(T(µi\|µ1, . . . , µL)− 1). Therefore from the assumption we have T(λ\|µ1, . . . , µL)\|P̄ 〉 = f(λ)\|P̄ 〉 for some scalar f(λ). Taking the “column sum” on the both sides using the sum-to-unity property of T(λ\|µ1, . . . , µL) we find f(λ) = 1. In the sequel, we assume i = L without losing generality in the light of cyclicity of trace. Now the equality \|P̄ 〉 = T(µL\|µ1, . . . , µL)\|P̄ 〉 to be shown takes the form Tr(Xα1(µ1) · · ·XαL(µL)) = ∑ β1,...,βL T(µL\|µ1, . . . , µL)α1,...,αL β1,...,βL Tr(Xβ1(µ1) · · ·XβL(µL)). In the l.h.s., starting from XαL−1(µL−1)XαL(µL) apply the relation (4.5) repeatedly to send XαL(µL) to the left through the whole product to bring it back to the original rightmost position by using the cyclicity of the trace. It results in the relation Tr(Xα1(µ1) · · ·XαL(µL)) = ∑ β1,...,βL M ( βL\|α1,...,αL−1 β1,...,βL−1 \|αL ) Tr(Xβ1(µ1) · · ·XβL(µL)) in terms of M defined under (2.20). It is depicted as (2.18). Since the parameters attached to the horizontal arrow and the rightmost vertical arrow are both µL, we have M ( βL\|α1,...,αL−1 β1,...,βL−1 \|αL ) = T(µL\|µ1, . . . , µL)α1,...,αL β1,...,βL from the right relation in (2.15). � The sum-to-unity property assures dim Ker(T(µi\|µ1, . . . , µL)− 1) ≥ 1 in general. We expect that this is 1 when µ1, . . . , µL are distinct8. Anyway the matrix product formula (4.4) has been proved in [34, Proposition 6] without relying on the assumption dim Ker(T(µi\|µ1, . . . , µL)−1) = 1 but using the auxiliary condition [34, equation (30)] of S(λ, µ). We have included Proposition 4.3 as a possible variant avoiding such a specific property of the stochastic R matrix. 8It is greater than 1 for example when µ1 = · · · = µL. 14 A. Kuniba, M. Okado and S. Watanabe 5 Zamolodchikov–Faddeev algebra 5.1 General remarks Let us write (4.5) symbolically as X(µ)⊗X(λ) = Š(λ, µ) [ X(λ)⊗X(µ) ] , (5.1) where X(µ) = (Xα(µ)) denotes a collection of operators. The quadratic relation of this form is called Zamolodchikov–Faddeev (ZF) algebra. Its associativity is guaranteed by the YBE satisfied by the structure function Š(λ, µ)9. Before presenting the specific results to our ZRP in the next subsection, we review its background briefly in this subsection. Similar contents can also be found for example in [10, 14, 20, 41] and references therein. ZF algebra was originally introduced in the integrable quantum field theory in (1 + 1) di- mension to encode the factorized scattering of particles [18, 51]. The structure function therein should be a properly normalized scattering matrix satisfying unitarity to guarantee the total probability conservation in the quantum field theoretical setting. In the realm of integrable Markov processes, the situation is parallel. The ZF algebra serves as a local version of the stationary condition in the matrix product construction of the stationary states. The structure function, stochastic R matrix, should fulfill the sum-to-unity property. It was demonstrated in the proof of Proposition 4.3 how the ZF algebra leads to the stationary condition of the system in the case of a discrete time Markov process. Historically, however, such quadratic relations were utilized earlier in continuous time models as ‘cancellation mechanism’ or ‘hat relations’ [15]. In the present set-up it reads h [ X(µ)⊗X(µ) ] = X(µ)⊗X ′(µ)−X ′(µ)⊗X(µ) (5.2) with h = ∂ ∂λ Š(λ, µ)\|λ=µ. This is the derivative of (5.1) at λ = µ with the additional condition Š(µ, µ) = id which matches (2.15). In terms of components, it reads ∑ γ,δ hα,βγ,δXγ(µ)Xδ(µ) = Xα(µ)X ′β(µ)−X ′α(µ)Xβ(µ) if the action of h is set as h(\|α〉 ⊗ \|β〉) = ∑ γ,δ hγ,δα,β\|γ〉 ⊗ \|δ〉. The rela- tion (5.2) manifestly tells that the matrix product states Tr(X(µ)⊗ · · ·⊗X(µ)) are null vectors of the operator H = ∑ i∈ZL hi,i+1 in the periodic setting. In retrospect, one may compare the relation between (5.2) and (5.1) with that between the XXZ model and the six-vertex model in the light of Baxter’s formula (3.5). Back to the ZF algebra itself, it is naturally embedded into the so-called RLL = LLR relation[ L(λ)⊗ L(µ) ] Š(λ, µ) = Š(λ, µ) [ L(λ)⊗ L(µ) ] (5.3) for an L operator L(λ) if there is a special index, say 0, such that S(λ, µ)β,α0,0 = S(λ, µ)0,0β,α = θ(α = β = 0). In fact the matrix element of (5.3) for \|0〉 ⊗ \|0〉 → \|α〉 ⊗ \|β〉 gives (5.1) by the identification L(µ)α,0 = Xα(µ). In view of this, construction of stationary states is elevated and embedded into that of representations of the stochastic RLL = LLR relation in which the relevant components of Tr(L(µ1)⊗ · · · ⊗ L(µL)) are convergent and not identically zero. When the structure function is the R matrix R1,1 of the vector representation, a universal L has been provided in [23]. Starting from it, one can cope with RLL = LLR with higher Rl,m by the corresponding fusion of L’s in principle [28]. Modifying it so as to fit the stochastic Sl,m should be feasible by an appropriate twist (cf. [20]). In this sense there is a standard route to achieve the matrix product construction for Sl,m-based models at least conceptually if not practically. On the other hand, a further intriguing feature is expected when the structure function is S(λ, µ) 9Š(λ, µ) = PS(λ, µ) with P (\|α〉 ⊗ \|β〉) = \|β〉 ⊗ \|α〉 so that S(λ, µ)β,αγ,δ = Š(λ, µ)α,βγ,δ . Integrable Structure of Multispecies Zero Range Process 15 due to the peculiarity of its origin (2.8)–(2.11). This is one of our motivations in Sections 5.2 and 5.3. Turning to stationary probabilities, the maneuver in the proof of Proposition 4.3 eluci- dates that it is a part of more general problem of finding solutions to a quantum Knizhnik– Zamolodchikov type equation [19] with appropriate subsidiary conditions. Such wider problems have not been addressed for our stochastic R matrix (2.11). Implication of the sum-to-unity property (2.13) to the ZF algebra will be explained after Remark 5.3 until the end of Section 5. In the next subsection we will be concerned with a particular representation of the ZF algebra in terms of q-bosons. 5.2 q-boson representation Let us present a q-boson representation of the ZF algebra (4.5)10. Here and in the next subsection we stay in the regime 0 < q < 1. From (2.11) it has the explicit form Xα(µ)Xβ(λ) = ∑ γ≤α Φq(β\|α+ β − γ;λ, µ)Xγ(λ)Xα+β−γ(µ), (5.4) where the omitted condition γ ∈ Zn≥0 should always be taken for granted. We find it convenient to work also with another normalization Zα(µ) specified by Xα(µ) = gα(µ)Zα(µ), (5.5) where gα(µ) has been defined in (4.3). The ZF algebra for the latter takes the form Zα(µ)Zβ(λ) = ∑ γ≤α qϕ(α−γ,β−γ)Φq(γ\|α;λ, µ)Zγ(λ)Zα+β−γ(µ) (5.6) due to the identity gγ(λ)gα+β−γ(µ) gα(µ)gβ(λ) Φq(β\|α+ β − γ;λ, µ) = qϕ(α−γ,β−γ)Φq(γ\|α;λ, µ). Let B be the algebra generated by 1, b, c, k obeying the relations kb = qbk, kc = q−1kc, bc = 1− k, cb = 1− qk. (5.7) We call it the q-boson algebra. It has a basis {bicj \| i, j ∈ Z≥0}. Let F = ⊕ m≥0C(q)\|m〉 be the Fock space and F ∗ = ⊕ m≥0C(q)〈m\| be its dual on which the q-boson operators b, c, k act as b\|m〉 = \|m+ 1〉, c\|m〉 = (1− qm)\|m− 1〉, k\|m〉 = qm\|m〉, 〈m\|c = 〈m+ 1\|, 〈m\|b = 〈m− 1\|(1− qm), 〈m\|k = 〈m\|qm, (5.8) where \|−1〉 = 〈−1\| = 0 and 1 acts as the identity. They satisfy the defining relations (5.7). The bilinear pairing of F ∗ and F is specified as 〈m\|m′〉 = δm,m′(q)m. Then 〈m\|(X\|m′〉) = (〈m\|X)\|m′〉 is valid and the trace is given by Tr(X) = ∑ m≥0 〈m\|X\|m〉 (q)m . As a vector space, the q-boson algebra B has the direct sum decomposition B = C(q)1 ⊕ Bfin, where Bfin = ⊕ r≥1(Br+ ⊕ Br− ⊕ Br0) with Br+ = ⊕ s≥0C(q)ksbr, Br− = ⊕ s≥0C(q)kscr and Br0 = C(q)kr. The trace Tr(X) is convergent if X ∈ Bfin. It vanishes unless X ∈ ⊕ r≥1 Br0 when it is evaluated by Tr(kr) = (1− qr)−1. 10In this paper we do not attempt to classify the representations such that r.h.s. of (4.4) is finite and nonzero. 16 A. Kuniba, M. Okado and S. Watanabe By the q-boson representation we mean the algebra homomorphisms ZF algebra (5.4) or (5.6)→ B⊗n(n−1)/2 → End ( F⊗n(n−1)/2 ) . The right arrow is already given by (5.8). In the sequel we will focus on the left arrow and denote Zα(µ) 7→ (· · · ) simply by Zα(µ) = (· · · ). The ZF algebra (5.6) admits a “trivial” representation Zα(ζ) = Kα in terms of an opera- tor Kα satisfying K0 = 1 and KαKβ = qϕ(α,β)Kα+β [34, Proposition 7], where ϕ(α, β) is defined in (2.9). Such a Kα is easily constructed, for instance as11 Kα1,...,αn = kα + 1 cα1 ⊗ · · · ⊗ kα + n−1cαn−1 ∈ B⊗n−1, α+ i := αi+1 + · · ·+ αn. (5.9) However this representation does not contain a creation operator b therefore leads to vanish- ing trace in the matrix product formula (4.4). Our Zα(ζ) given below may be regarded as a perturbation series starting from the trivial representation in terms of creation operators. For αi ∈ Z≥0, define the element Zα1,...,αn(ζ) ∈ B⊗n(n−1)/2 from the n = 1 case and the recursion with respect to n as follows: Zα1(ζ) = 1, (5.10) Zα1,...,αn(ζ) = ∑ l=(l1,...,ln−1)∈Zn−1 ≥0 Xl(ζ)⊗ bl1kα + 1 cα1 ⊗ · · · ⊗ bln−1kα + n−1cαn−1 , (5.11) where Xl(ζ) = gl(ζ)Zl(ζ) as in (5.5) and α+ i is defined by (5.9). Theorem 5.1 ([35]). The Zα(ζ) defined by (5.10), (5.11) satisfies the ZF algebra (5.6) for general n. 5.3 Explicit formula From (5.11) Zα(ζ) depends on α simply as Zα1,...,αn(ζ) = Z0n(ζ) ( 1⊗ 1 2 (n−1)(n−2) ⊗Kα1,...,αn ) , (5.12) in terms of Kα1,...,αn in (5.9). Thus it suffices to calculate the special case of (5.11): Z0n(ζ) = ∑ l1,...,ln−1∈Z≥0 gl1,...,ln−1(ζ)Zl1,...,ln−1(ζ)⊗ bl1 ⊗ · · · ⊗ bln−1 . (5.13) Utilizing (zw)∞ (z)∞ = ∑ j≥0 (w)j (q)j zj , we get the explicit formula for n = 2: Z0,0(ζ) = ∑ l1≥0 (ζ)l1ζ −l1 (q)l1 bl1 = (b)∞ (ζ−1b)∞ , Zα1,α2(ζ) = Z0,0(ζ)Kα1,α2 = (b)∞ (ζ−1b)∞ kα2cα1 , Xα1,α2(ζ) = gα1,α2(ζ)Zα1,α2(ζ) = ζ−α1−α2(ζ)α1+α2 (q)α1(q)α2 (b)∞ (ζ−1b)∞ kα2cα1 . (5.14) It is a good exercise to confirm Example 4.1 by substituting these results into the matrix product formula (4.4). 11We write Kα with α = (α1, . . . , αn) as Kα1,...,αn rather than K(α1,...,αn) for simplicity. A similar convention will also be used for gα(ζ), Xα(ζ) and Zα(ζ). Integrable Structure of Multispecies Zero Range Process 17 For n = 3, the sum (5.13) is calculated by using (ζ)l1+l2 = (ζ)l2(ql2ζ)l1 as Z0,0,0(ζ) = ∑ l1,l2 ζ−l1−l2(ζ)l1+l2 (q)l1(q)l2 (b)∞ (ζ−1b)∞ kl2cl1 ⊗ bl1 ⊗ bl2 = (b⊗ 1⊗ 1)∞ (ζ−1b⊗ 1⊗ 1)∞ ∑ l2 ζ−l2(ζ)l2(k⊗ 1⊗ b)l2 (q)l2 ∑ l1 ζ−l1(ql2ζ)l1(c⊗ b⊗ 1)l1 (q)l1 = (b⊗ 1⊗ 1)∞ (ζ−1b⊗ 1⊗ 1)∞ ∑ l2 ζ−l2(ζ)l2(k⊗ 1⊗ b)l2 (q)l2 (ql2c⊗ b⊗ 1)∞ (ζ−1c⊗ b⊗ 1)∞ = (b⊗ 1⊗ 1)∞ (ζ−1b⊗ 1⊗ 1)∞ (c⊗ b⊗ 1)∞ ∑ l2 ζ−l2(ζ)l2(k⊗ 1⊗ b)l2 (q)l2 1 (ζ−1c⊗ b⊗ 1)∞ = (b⊗ 1⊗ 1)∞ (ζ−1b⊗ 1⊗ 1)∞ (c⊗ b⊗ 1)∞ (k⊗ 1⊗ b)∞ (ζ−1k⊗ 1⊗ b)∞ 1 (ζ−1c⊗ b⊗ 1)∞ , Zα1,α2,α3(ζ) = Z0,0,0(ζ)(1⊗Kα1,α2,α3) = Z0,0,0(ζ)(1⊗ kα2+α3cα1 ⊗ kα3cα2). These results on Z0,0(ζ) and Z0,0,0(ζ) are summarized as12 Z0,0(ζ) = V1(1)V1(ζ)−1, V1(ζ) = (ζ−1b)∞, Z0,0,0(ζ) = ( Z0,0(ζ)⊗ 1⊗ 1 ) V2(1)V2(ζ)−1, V2(ζ) = (ζ−1c⊗ b⊗ 1)∞(ζ−1k⊗ 1⊗ b)∞. (5.15) Now we proceed to general n ≥ 2 case. Substitution of (5.12)\|n→n−1 into the r.h.s. of (5.13) gives Z0n(ζ) = ( Z0n−1(ζ)⊗ 1⊗n−1 ) Yn(ζ), Yn(ζ) = ∑ l1,...,ln−1∈Z≥0 gl1,...,ln−1(ζ)1⊗ 1 2 (n−2)(n−3) ⊗Kl1,...,ln−1 ⊗ bl1 ⊗ · · · ⊗ bln−1 . (5.16) It is handy to describe B⊗n(n−1)/2 by introducing the copies Bi,j = 〈1,bi,j , ci,j ,ki,j〉 of the q-boson algebras for 1 ≤ i ≤ j < n obeying (5.7) within each Bi,j and [Bi,j ,Bi′,j′ ] = 0 if (i, j) 6= (i′, j′). We take them so that Zα1,...,αn(ζ) ∈ ⊗ 1≤i≤j<n Bi,j and (5.16) reads Yn(ζ) = ∑ l1,...,ln−1∈Z≥0 gl1,...,ln−1(ζ) ( k l+1 1,n−2c l1 1,n−2 · · ·k l+n−2 n−2,n−2c ln−2 n−2,n−2 )( bl11,n−1 · · ·b ln−1 n−1,n−1 ) , where l+j = lj+1 + · · ·+ ln−1. It corresponds to labeling the components in B⊗n(n−1)/2 as (1, 1), (1, 2), (2, 2), (1, 3), (2, 3), (3, 3), . . . , (1, n− 1), (2, n− 1), . . . , (n− 1, n− 1). (5.17) Define the following elements in ⊗ 1≤i≤j<n Bi,j (actually in a certain completion of it): Yj(ζ) = Vj−1(1)Vj−1(ζ)−1 = Vj−1(ζ)−1Vj−1(1), Vj(ζ) = (ζ−1A1,j)∞(ζ−1A2,j)∞ · · · (ζ−1Aj,j)∞, Ai,j = k1,j−1k2,j−1 · · ·ki−1,j−1ci,j−1bi,j , cj,j−1 = 1. In particular, A1,n−1 = c1,n−2b1,n−1 and An−1,n−1 = k1,n−2 · · ·kn−2,n−2bn−1,n−1. As for the right equality in the first line, see Remark 5.3. 12The operators Vj(ζ)’s appearing in the sequel have nothing to do with that in (2.1). 18 A. Kuniba, M. Okado and S. Watanabe Theorem 5.2 ([35]). The representation of the ZF algebra (5.6) in ⊗ 1≤i≤j<n Bi,j given in Theorem 5.1 and (5.10), (5.11) is expressed as follows: Zα1,...,αn(ζ) = Z0n(ζ)k α+ 1 1,n−1c α1 1,n−1 · · ·k α+ n−1 n−1,n−1c αn−1 n−1,n−1, α+ i = αi+1 + · · ·+ αn, Z0n(ζ) = Y2(ζ)Y3(ζ) · · ·Yn(ζ). The cases n = 2, 3 reproduce (5.15) under the identification x1,1 = x⊗1⊗1, x1,2 = 1⊗x⊗1, x2,2 = 1 ⊗ 1 ⊗ x in accordance with (5.17). The recursive construction (5.13) with respect to rank may be viewed as reminiscent of nested Bethe ansatz. A similar structure has also been observed in multispecies ASEP [10, 40]. It is not hard to show that the substitution of the formulas in Theorem 5.2 and (5.5) into the matrix product formula (4.4) of P(σ1, . . . , σL) leads to the convergent trace provided that the configuration (σ1, . . . , σL) belongs to a basic sector explained in Section 4.1. Remark 5.3. As a corollary of the ZF algebra (4.5) and S(λ, µ)0,0γ,δ = θ(γ = δ = 0) one can derive [Z0(µ), Z0(λ)] = 0, [Vm(µ), Vm(λ)] = 0, 1 ≤ m ≤ n− 1. Let us comment on the implication of the sum-to-unity property (2.13) to the ZF algebra (4.5). Using the weight conservation it implies [A(µ\|w), A(λ\|w)] = 0 for A(λ\|w) = ∑ α Xα(λ)wα, (5.18) where wα = wα1 1 · · ·wαnn . Example 5.4. The generating function in (5.18) for n = 2 and w = (w1, w2) = (x, y) can be computed as A(λ\|w) = (b)∞ (λ−1b)∞ ∑ l,m≥0 xmylλ−l−m(λ)l+m (q)l(q)m klcm = (b)∞( λ−1b ) ∞ ∑ m≥0 (xλ−1)m(λ)m (q)m (qmyk)∞ (yλ−1k)∞ cm = (b)∞ (λ−1b)∞ Γ ( xλ−1, yλ−1 )−1 Γ(x, y), Γ(x, y) = (xc)∞(yk)∞. It plays an important role to investigate the stationary states in ‘grand canonical ensemble’ picture (cf. [16]). In the matrix product formula (4.4), multiply wσ1+···+σL and take the sum over the states (σ1, . . . , σL) ∈ (Zn≥0)L belonging to all the basic sectors. It yields the series Tr(A(µ1\|w) · · ·A(µL\|w))′ = ∑ α1+···+αL∈Zn≥1 wα1+···+αL Tr(Xα1(µ1) · · ·XαL(µL)) = ∑ k∈Zn≥1 wkGk(µ1, . . . , µL; q), where the prime drops the terms corresponding to α1+ · · ·+αL ∈ Zn≥0 \Zn≥1 to take into account the basic sectors only. We have introduced the function Gk(µ1, . . . , µL; q) = ∑ α1+···+αL=k Tr(Xα1(µ1) · · ·XαL(µL)). Integrable Structure of Multispecies Zero Range Process 19 This is the normalization constant of the stationary state in the basic sector B(k) (4.2). The com- mutativity in (5.18) implies that Gk(µ1, . . . , µL; q) is symmetric function in µ1, . . . , µL. Moreover from (4.3), (5.5), Theorem 5.2 and the expansion formula (zw)∞ (z)∞ = ∑ j≥0 (w)j (q)j zj , it follows that Gk(µ1, . . . , µL; q) is a polynomial in µ−1i ’s. It is rational in q because the trace is evaluated as Tr(kr) = (1 − qr)−1. Detailed study of Gk(µ1, . . . , µL; q) and the related Conjecture 4.2 is a future problem13. Similar observations in such a direction have led to matrix product formulas for Macdonald polynomials and their generalizations for some other integrable lattice models. See for example [8, 10, 11, 20, 38]. 6 R matrices of generalized quantum group We have seen that the factorization (2.8) led to significant consequences in previous sections. Here we generalize it to a part of 2n+1 quantum R matrices labeled with (ε1, . . . , εn+1) ∈ {0, 1}n+1. The one treated so far corresponds to the choice (ε1, . . . , εn+1) = (0, . . . , 0). These R matrices have been obtained from the special solutions to the tetrahedron equation by a certain reduction [36]. The underlying algebra has been identified with a generalized quantum group. We shall present these results with a brief background based on [36]. 6.1 Definition of UA In this subsection we assume that n is a positive integer. Let (ε1, . . . , εn+1) be a sequence of 0 or 1. In what follows the indices i, j are understood to be elements in Zn+1. We write i ≡ j to mean i = j in Zn+1. For i, j = 0, 1, . . . , n ∈ Zn+1, set qi = { q, εi = 0, −q−1, εi = 1, Dij = Dji =  qiqi+1, j ≡ i, q−1i , j ≡ i− 1, q−1i+1, j ≡ i+ 1, 1, otherwise. Let UA = UA(ε1, . . . , εn+1) be a Q(q)-algebra generated by ei,fi, k ±1 i (i ∈ Zn+1) obeying the following relations. (We use the notation [u] = (qu − q−u)/(q − q−1).) kik −1 i = k−1i ki = 1, kikj = kjki, kiejk −1 i = Dijej , kifjk −1 i = D−1ij fj , (6.1) [ei, fj ] = δij ki − k−1i q − q−1 , (6.2) e2i = f2i = 0 if εi 6= εi+1, (6.3) [ei, ej ] = [fi, fj ] = 0 if j 6≡ i, i± 1, (6.4) e2i ej − (−1)εi [2]eiejei + eje 2 i = (e→ f) = 0 if εi = εi+1, j ≡ i± 1, (6.5) eiei−1eiei+1 + (−1)εi [2]eiei−1ei+1ei − eiei+1eiei−1 − ei−1eiei+1ei + ei+1eiei−1ei = (e→ f) = 0 if εi 6= εi+1. (6.6) The algebra UA with the relations (6.1) and (6.2) was introduced for n ≥ 1 in [36] as a sym- metry algebra characterizing solutions to the YBE obtained by the 2D reduction procedure to be explained in Section 6.2 from the tetrahedron equation [50]. Based on the observation in [36, Section 3.3], the relations (6.3)–(6.6) were supplemented when n ≥ 2 [37] by showing that the 13We expect that the condition k ∈ Zn≥1 can slightly be relaxed in view of the convergence of Tr(Xα1(µ1) · · ·XαL(µL)) in the non-basic sector (k1, k2) with k1 = 0, k2 ≥ 1 for n = 2. See (5.14). 20 A. Kuniba, M. Okado and S. Watanabe subalgebra generated by ei, fi, ki for i = 1, . . . , n is isomorphic, up to adding simple generators, to the quantized universal enveloping super algebra of type A [48], where they correspond to a q-analogue of the Serre relations. The forthcoming construction (6.19) and all the subsequent claims are valid for n ≥ 1. UA is a Hopf algebra with coproduct ∆ given by ∆k±1i = k±1i ⊗ k ±1 i , ∆ei = 1⊗ ei + ei ⊗ ki, ∆fi = fi ⊗ 1 + k−1i ⊗ fi. (6.7) For the counit and the antipode, see [36, equation (3.4)]. To present a representation of UA, we introduce the following vector spaces: F = W (0) = ⊕ m≥0 C\|m〉(0), V = W (1) = C\|0〉(1) ⊕ C\|1〉(1), (6.8) W = W (ε1) ⊗ · · · ⊗W (εn+1) = ⊕ α1,...,αn+1 C\|α1, . . . , αn+1〉, (6.9) \|α1, . . . , αn+1〉 = \|α1〉(ε1) ⊗ · · · ⊗ \|αn+1〉(εn+1), Vl = ⊕ α,\|α\|=l C\|α〉 ⊂ W. (6.10) Note that the range of the indices αi are to be understood as Z≥0 or {0, 1} according to εi = 0 or 1, respectively. In (6.10) we have written \|α1, · · · , αn+1〉 simply as \|α〉. Proposition 6.1 ([36]). Let x be a parameter. The map πlx : UA → End(Vl) defined by πlx(ei)\|α〉 = xδi,0 [αi]\|α− ei + ei+1〉, πlx(fi)\|α〉 = x−δi,0 [αi+1]\|α+ ei − ei+1〉, πlx(ki)\|α〉 = (qi) −αi(qi+1) αi+1 \|α〉 (6.11) for 0 ≤ i ≤ n gives an irreducible representation when 0 ≤ l ≤ n + 1 if ε1 = · · · = εn+1 = 1, and l ∈ Z≥0 otherwise. The index i should be understood mod n + 1. In the right hand side of (6.11), vectors \|α′〉 = \|α′1, . . . , α′n+1〉 are to be understood as zero unless αi ∈ {0, 1} (εi = 1) and αi ∈ Z≥0 (εi = 0) for all 1 ≤ i ≤ n+ 1. Consider the following linear equation on R ∈ End(Vl ⊗ Vm):( πlx ⊗ πmy ) ∆op(g)R = R ( πlx ⊗ πmy ) ∆(g) ∀ g ∈ UA, (6.12) where ∆op(g) = P ◦∆ ◦ P with P (u⊗ v) = v ⊗ u. The dimension of the solution space to this equation is at most one if the UA-module Vl ⊗ Vm is irreducible. Conjecture 6.2 ([36]). The UA-module Vl ⊗ Vm is irreducible for any choice (ε1, . . . , εn+1) ∈ {0, 1}n+1. Theorem 6.3 ([36]). Conjecture 6.2 is true for (ε1, . . . , εn+1) = (1κ, 0n+1−κ) with 0 ≤ κ ≤ n+1. Suppose (ε1, . . . , εn+1) = (1κ, 0n+1−κ) with 0 ≤ κ ≤ n + 1. Then a little inspection on the representation (6.11) tells that the solution to (6.12) depends on x, y as R = R(z) with z = x/y. It will be denoted by R(z) = Rl,m(z) and referred to as quantum R matrix up to overall normalization. To fix the normalization, introduce ei = ( i−1︷︸︸︷ 0, . . . , 0, 1, n+1−i︷︸︸︷ 0, . . . , 0) ∈ Zn+1, e>m = em+1 + · · ·+ en+1, 1 ≤ m ≤ n+ 1. (6.13) Integrable Structure of Multispecies Zero Range Process 21 If ε1 = · · · = εn+1 = 1, we normalize Rl,m(z) (0 ≤ l,m ≤ n+ 1) as Rl,m(z)(\|e>n+1−l〉 ⊗ \|e>n+1−m〉) = \|e>n+1−l〉 ⊗ \|e>n+1−m〉. (6.14) If ε1 · · · εn+1 = 0, pick any i such that εi = 0 and normalize it as Rl,m(z)(\|lei〉 ⊗ \|mei〉) = \|lei〉 ⊗ \|mei〉. (6.15) In view of Conjecture 6.2, we expect that the quantum R matrix is characterized similarly for any (ε1, . . . , εn+1) ∈ {0, 1}n+1. 6.2 Construction of R matrix We briefly review how we constructed in [36] solutions depending on (ε1, . . . , εn+1) ∈ {0, 1}n+1 to the YBE from the tetrahedron equation. Define the 3D R operator R ∈ End ( W (0)⊗W (0)⊗W (0) ) by R ( \|i〉(0) ⊗ \|j〉(0) ⊗ \|k〉(0) ) = ∑ a,b,c∈Z≥0 R a,b,c i,j,k \|a〉 (0) ⊗ \|b〉(0) ⊗ \|c〉(0), where R a,b,c i,j,k is the one in (2.4). Similarly define the 3D L operator L ∈ End ( W (1) ⊗W (1) ⊗ W (0) ) [4] by L(\|i〉(1) ⊗ \|j〉(1) ⊗ \|k〉(0)) = ∑ a,b∈{0,1}, c∈Z≥0 L a,b,c i,j,k \|a〉 (1) ⊗ \|b〉(1) ⊗ \|c〉(0), L 0,0,j 0,0,m = L 1,1,j 1,1,m = δjm, L 0,1,j 0,1,m = −δjmqm+1, L 1,0,j 1,0,m = δjmq m, L 0,1,j 1,0,m = δjm−1 ( 1− q2m ) , L 1,0,j 0,1,m = δjm+1. It may be viewed as a six-vertex model with q-boson valued Boltzmann weights in the third component. Assign a solid arrow to F and a dotted arrow to V , and depict the matrix elements of 3D R and 3D L as To treat R and L on an equal footing we set M(0) = R and M(1) = L so that M(ε) ∈ End ( W (ε) ⊗W (ε) ⊗ F ) . They satisfy the following type of tetrahedron equation [4, 25, 36]14. M (ε) 1,2,4M (ε) 1,3,5M (ε) 2,3,6R4,5,6 = R4,5,6M (ε) 2,3,6M (ε) 1,3,5M (ε) 1,2,4. (6.16) This is an equality in End ( W (ε)⊗W (ε)⊗W (ε)⊗F ⊗F ⊗F ) . Subscripts of M (ε) i,j,k or Ri,j,k signify that they act on the i, j and k-th components of W (ε) ⊗W (ε) ⊗W (ε) ⊗ F ⊗ F ⊗ F , and do as the identity on the other. The tetrahedron equation (6.16) with ε = 1 is depicted as follows: 14M(0), M(1) were denoted by S(0), S(1) in [36]. 22 A. Kuniba, M. Okado and S. Watanabe The equation (6.16) with ε = 0 is expressed similarly by replacing all the dotted arrows by solid ones. Regarding (6.16) as a one-layer relation, we extend it to the (n+ 1)-layer version. Let ai W (εi), bi W (εi), ci W (εi) be copies of W (εi), where ai, bi and ci (i = 1, . . . , n + 1) are just distinct labels. Repeated use of (6.16) (n+ 1) times leads to( M (ε1) a1,b1,4 M (ε1) a1,c1,5 M (ε1) b1,c1,6 ) · · · ( M (εn+1) an+1,bn+1,4 M (εn+1) an+1,cn+1,5 M (εn+1) bn+1,cn+1,6 ) R4,5,6 = R4,5,6 ( M (ε1) b1,c1,6 M (ε1) a1,c1,5 M (ε1) a1,b1,4 ) · · · ( M (εn+1) bn+1,cn+1,6 M (εn+1) an+1,cn+1,5 M (εn+1) an+1,bn+1,4 ) . (6.17) This is an equality in End ( a W ⊗ b W ⊗ c W ⊗ 4 F ⊗ 5 F ⊗ 6 F ) , where a = (a1, . . . , an+1) is the array of labels and a W = a1 W (ε1) ⊗ · · · ⊗ an+1 W (εn+1). The notation b W and c W should be understood similarly. They are just copies of W defined in (6.9). One can reduce (6.17) to the YBE by evaluating the auxiliary space 4 F ⊗ 5 F ⊗ 6 F away appropriately. A natural way is to take trace of (6.17) over the auxiliary space after multiplying it with xh4(xy)h5yh6 from the left and R−14,5,6 from the right15. It results in the YBE Ra,b(x)Ra,c(xy)Rb,c(y) = Rb,c(y)Ra,c(xy)Ra,b(x) ∈ End ( a W ⊗ b W ⊗ c W ) (6.18) for the R matrix obtained as Ra,b(z) = ρ(z) Tr3 ( zh3M (ε1) a1,b1,3 · · ·M(εn+1) an+1,bn+1,3 ) ∈ End ( a W ⊗ b W ) , (6.19) where the scalar ρ(z) is inserted to control the normalization. The trace are taken with respect to the auxiliary Fock space F = 3 F signified by 3. Pictorially the matrix element (6.20) of (6.19) is expressed as follows: Here the broken arrows designate either F or V (6.8) according to εi = 0 or 1, and the winding arrow does 3 F over which the trace is taken. In short (6.19) is a matrix product construction of quantum R matrix R(z) by operators satisfying the tetrahedron equation. The formula (2.3) is just the concrete form of (6.19) for (ε1, . . . , εn+1) = (0, . . . , 0). It is not known if this type of construction extends much beyond the generalized quantum group UA. See [36, Section 2.8] for the list of the known results. However, the formula (6.19) is often more efficient than the fusion procedure practically. It also reveals a hidden 3D structure in a class of R matrices [4] and has led to another application to the multispecies totally asymmetric simple exclusion and zero range processes when ∀ εi = 1 and ∀ εi = 0 [30, 32]. Except for the two cases however, these R matrices do not satisfy the sum-to-unity in general16 and we have not found an application to stochastic systems. 15See around [36, equation (2.5)] for the definition of hi. 16This is partly because [29, Lemma 5] becomes trivial for s ≥ 3 invalidating the argument similar to the proof of [29, Theorem 6]. Integrable Structure of Multispecies Zero Range Process 23 Define the matrix elements of R(z) by R(z)(\|α〉 ⊗ \|β〉) = ∑ γ,δ R(z)γ,δα,β\|γ〉 ⊗ \|δ〉, (6.20) where \|α〉, . . . , \|δ〉 ∈ W (6.9). It satisfies R(z)γ,δα,β = 0 unless γ + δ = α+ β and \|γ\| = \|α\|, \|δ\| = \|β\|, which implies that R(z) decomposes into matrices acting on finite-dimensional vector spaces: R(z) = ⊕ l,m≥0 Rl,m(z), Rl,m(z) ∈ End(Vl ⊗ Vm), where the former sum ranges over 0 ≤ l,m ≤ n+ 1 if ε1 = · · · = εn+1 = 1 and l,m ∈ Z≥0 other- wise17. The normalization (6.14) and (6.15) is achieved by choosing ρ(z) = (−q)−max(m−l,0)(1− q\|l−m\|z) and ρ(z) = z−m(ql−mz;q2)m+1 (ql−m+2z−1;q2)m in (6.19), respectively [36, Section 2.6]. When l = m, we have Rm,m(1)(\|α〉 ⊗ \|β〉) = \|β〉 ⊗ \|α〉. Proposition 6.4 ([36]). For arbitrary sequence (ε1, . . . , εn+1) ∈ {0, 1}n+1, the matrix Rl,m(z = x/y) satisfies (6.12). From Theorem 6.3 and Proposition 6.4 it follows that Rl,m(z) associated with the sequence (ε1, . . . , εn+1) = (1κ, 0n+1−κ) with 0 ≤ κ ≤ n+ 1 is indeed the quantum R matrix of UA. Example 6.5. Consider UA(1, 0). For l,m ≥ 1, one has Vm = C\|0,m〉⊕C\|1,m−1〉 ⊂ W = V ⊗F and similarly for Vl. The action of R(z) on Vl ⊗ Vm is given by R(z)(\|0, l〉 ⊗ \|0,m〉) = \|0, l〉 ⊗ \|0,m〉, R(z)(\|1, l − 1〉 ⊗ \|0,m〉) = 1− q2m z − ql+m \|0, l〉 ⊗ \|1,m− 1〉+ qmz − ql z − ql+m \|1, l − 1〉 ⊗ \|0,m〉, R(z)(\|0, l〉 ⊗ \|1,m− 1〉) = qlz − qm z − ql+m \|0, l〉 ⊗ \|1,m− 1〉+ (1− q2l)z z − ql+m \|1, l − 1〉 ⊗ \|0,m〉, R(z)(\|1, l − 1〉 ⊗ \|1,m− 1〉) = 1− ql+mz z − ql+m \|1, l − 1〉 ⊗ \|1,m− 1〉. 6.3 Special value of R matrix In this subsection we wish to obtain an explicit form of R(z) = Rl,m(z) at z = ql−m. We are going to show the following theorem. Theorem 6.6. When l ≤ m and (ε1, . . . , εn+1) = (1κ, 0n+1−κ) with 0 ≤ κ ≤ n+ 1, the following formula is valid: R ( z = ql−m )γ,δ α,β = δγ+δα+βq ψ+l(l−m)θ(κ=n+1) ( m l )θ(κ=n+1)−1 q2 n+1∏ i=1 ( βi γi ) q2 , (6.21) ψ = ψγ,δα,β = ∑ 1≤i<j≤n+1 αi(βj − γj) + ∑ 1≤i<j≤n+1 (βi − γi)γj . 17The notation Vl matches (2.1) when ∀ εi = 0. It was denoted by Wl in [36] although. 24 A. Kuniba, M. Okado and S. Watanabe Set î = ei − ei+1 for i ∈ Zn+1. It is straightforward to check Lemma 6.7. ψγ−î,δα,β − ψγ,δ−îα,β = γi+1 − αi + βi − γi + 1 + (l −m)δi,0, ψγ,δ α+î,β − ψγ,δ−îα,β = βi+1 − γi+1 + (l −m)δi,0, ψγ,δ α,β+î − ψγ,δ−îα,β = γi+1 − αi. Proposition 6.8. Denote the r.h.s. of (6.21) by Xγ,δ α,β and set X(\|α〉 ⊗ \|β〉) = ∑ γ,δ Xγ,δ α,β\|γ〉 ⊗ \|δ〉. Suppose l ≤ m. Then for any x, y such that x/y = ql−m and (ε1, . . . , εn+1) ∈ {0, 1}n+1, we have( πlx ⊗ πmy ) ∆op(g)X = X ( πlx ⊗ πmy ) ∆(g) ∀ g ∈ UA(ε1, . . . , εn+1). (6.22) Proof. It suffices to show it for g = ki, ei, fi in the four cases (εi, εi+1) = (1, 1), (1, 0), (0, 1), (0, 0). The case (εi, εi+1) = (0, 0) was done in [29]. Here we treat (εi, εi+1) = (0, 1) case. The proof for (εi, εi+1) = (1, 1), (1, 0) are similar. The relation (6.22) with g = ki means the weight conservation and it holds due to the factor δγ+δα+β. In the sequel we show (6.22) for g = fi. The case g = ei is similar hence omitted. Let the both sides of (6.22) act on \|α〉 ⊗ \|β〉 ∈ Vl ⊗ Vm and compare the coefficients of \|γ〉 ⊗ \|δ〉 ∈ Vl ⊗ Vm in the output vector. Using (6.7), (6.11) and (6.20) we find that the relation to be proved is Xγ,δ−î α,β [δi+1 + 1]θ(δi+1 = 0) +Xγ−î,δ α,β qδi+δi+1(−1)−δi+1z−δi,0 [γi+1 + 1]θ(γi+1 = 0) = Xγ,δ α+î,β [αi+1]θ(αi+1 = 1)z−δi,0 +Xγ,δ α,β+î [βi+1]θ(βi+1 = 1)qαi+αi+1(−1)−αi+1 at z = ql−m under the weight conservation (i) αi + βi = γi + δi − 1 and (ii) αi+1 + βi+1 = γi+1 + δi+1 + 1. By substituting (6.21) and applying Lemma 6.7, it is simplified to [δi+1 + 1] ( 1− q2βi+1 )( 1− q2(βi−γi+1) )( 1− q2(γi+1+1) ) θ(δi+1 = 0) + qγi+1+δi+1+2βi−2γi+2[γi+1 + 1] ( 1− q2βi+1 )( 1− q2(βi+1−γi+1) ) × ( 1− q2γi ) (−1)δi+1θ(γi+1 = 0) = qβi+1−γi+1 [αi+1] ( 1− q2βi+1 )( 1− q2(βi−γi+1) )( 1− q2(γi+1+1) ) θ(αi+1 = 1) + qγi+1+αi+1 [βi+1] ( 1− q2(βi+1) )( 1− q2(βi+1−γi+1) )( 1− q2(γi+1+1) ) × (−1)αi+1θ(βi+1 = 1). From the weight conservation (ii) αi+1 + βi+1 = γi+1 + δi+1 + 1 and εi+1 = 1, we have only four cases (αi+1, βi+1, γi+1, δi+1) = (1, 1, 1, 0), (1, 0, 0, 0), (0, 1, 0, 0), (1, 1, 0, 1). They are easily checked by using (i). � Proof of Theorem 6.6. The UA-module Vl ⊗ Vm is irreducible for the choice (ε1, . . . , εn+1) = (1κ, 0n+1−κ) with 0 ≤ κ ≤ n + 1 by [36, Proposition 6.11]. (This fact has been quoted as Theorem 6.3 in this paper.) Therefore R(z) is uniquely characterized by the relation (6.22) up to normalization. The agreement of the normalization is readily checked. � If Conjecture 6.2 holds, Proposition 6.8 tells that the factorized formula in Theorem 6.6 is valid for arbitrary (ε1, . . . , εn+1) ∈ {0, 1}n+1. Integrable Structure of Multispecies Zero Range Process 25 6.4 Parameter version For n ∈ Z≥1 and ε = (ε1, . . . , εn) ∈ {0, 1}n, set B(ε) = {(α1, . . . , αn) \|αi ∈ {0, 1} if εi = 1, αi ∈ Z≥0 if εi = 0}, W (ε) = ⊕ (α1,...,αn)∈B(ε) C\|α1, . . . , αn〉. Note that we have shifted to the n-component setting. Introduce the operator S(ε)(λ, µ) ∈ End(W (ε)⊗W (ε)) depending on the parameters λ, µ by S(ε)(λ, µ)(\|α〉 ⊗ \|β〉) = ∑ γ,δ∈B(ε) S(λ, µ)γ,δα,β\|γ〉 ⊗ \|δ〉, (6.23) where the element S(λ, µ)γ,δα,β is specified by exactly the same formula as (2.11) and (2.9). In other words S(ε)(λ, µ) is a restriction of S(λ, µ) ∈ End(W ⊗W ) on W (ε)⊗W (ε), where W was defined before (2.10). It corresponds to the parameter version of R(z = ql−m) for (ε1, . . . , εn+1) with εn+1 = 0.18 Combining the YBE (6.18), Theorem 6.6 and the argument similar to [29, Section 2.3], one can generalize (2.12) to Theorem 6.9. For ε = (1κ, 0n−κ) with 0 ≤ κ ≤ n, S(ε)(λ, µ) satisfies the YBE: S (ε) 1,2(ν1, ν2)S (ε) 1,3(ν1, ν3)S (ε) 2,3(ν2, ν3) = S (ε) 2,3(ν2, ν3)S (ε) 1,3(ν1, ν3)S (ε) 1,2(ν1, ν2). In view of Conjecture 6.2 we also conjecture that the above YBE is valid for arbitrary ε ∈ {0, 1}n. A direct proof of this assertion will not be difficult although we do not pursue it here. On the other hand, the sum-to-unity (2.13) does not hold in general if ε 6= (0, . . . , 0). Here is the simplest example. Example 6.10. S(ε)(λ, µ) with n = 1 and ε = (1) defines a five vertex model whose vertex weights read in the convention (2.14). One sees that the sum-to-unity (2.13) is invalid when (α, β) = (1, 1) because S(1)(λ, µ)0,21,1 = λ−µ λ(1−µ) must be dismissed from the sum. These weights also possess the NW-freeness mentioned in Remark 3.2. 7 Summary We have reviewed the construction of the multispecies ZRP in [29], the matrix product for- mula for the stationary probability in [34, 35] and the relevant quantum R matrices originat- ing in the tetrahedron equation and the generalized quantum groups in [36]. We have also pointed out a new commuting Markov transfer matrix in Section 2.3, the associated Markov Hamiltonian (3.10), the Serre type relations (6.3)–(6.6) for the generalized quantum group UA(ε1, . . . , εn+1), factorization of its quantum R matrix at special point of the spectral parameter in Theorem 6.6, and the parameter version of the R matrix (6.23) satisfying the YBE. 18For example S(1,...,1)(λ, µ) originates in Rl,m(z) with ‘inhomogeneous’ choice (ε1, . . . , εn+1) = (1, . . . , 1, 0). 26 A. Kuniba, M. Okado and S. Watanabe A Example of Rl,m(z) for UA(1, 1, 0) Consider UA(1, 1, 0). For l,m ≥ 2, one has Vm = C\|0, 0,m〉 ⊕ C\|0, 1,m − 1〉 ⊕ C\|1, 0,m − 1〉 ⊕ C\|1, 1,m − 2〉 ⊂ W = V ⊗ V ⊗ F and similarly for Vl. The action of R(z) on Vl ⊗ Vm is given by R(z)(\|0, 0, l〉 ⊗ \|0, 0,m〉) = \|0, 0, l〉 ⊗ \|0, 0,m〉, R(z)(\|0, 0, l〉 ⊗ \|0, 1,m− 1〉) = qlz − qm z − ql+m \|0, 0, l〉 ⊗ \|0, 1,m− 1〉 + (1− q2l)z z − ql+m \|0, 1, l − 1〉 ⊗ \|0, 0,m〉, R(z)(\|0, 0, l〉 ⊗ \|1, 0,m− 1〉) = qlz − qm z − ql+m \|0, 0, l〉 ⊗ \|1, 0,m− 1〉 + (1− q2l)z z − ql+m \|1, 0, l − 1〉 ⊗ \|0, 0,m〉, R(z)(\|0, 0, l〉 ⊗ \|1, 1,m− 2〉) = (qm − qlz)(qm − ql+2z) (ql+m − z)(ql+m − q2z) \|0, 0, l〉 ⊗ \|1, 1,m− 2〉 + (qlz − qm)(1− q2l)zq2 (ql+m − z)(ql+m − q2z) \|0, 1, l − 1〉 ⊗ \|1, 0,m− 1〉 + (qlz − qm)(1− q2l)zq (ql+m − z)(ql+m − q2z) \|1, 0, l − 1〉 ⊗ \|0, 1,m− 1〉 + (1− q2l)(1− q2l−2)z2q2 (ql+m − z)(ql+m − q2z) \|1, 1, l − 2〉 ⊗ \|0, 0,m〉, R(z)(\|1, 1, l − 2〉 ⊗ \|0, 0,m〉) = q2(1− q2m)(1− q2m−2) (ql+m − z)(ql+m − q2z) \|0, 0, l〉 ⊗ \|1, 1,m− 2〉 + q2(1− q2m)(qmz − ql) (ql+m − z)(ql+m − q2z) \|0, 1, l − 1〉 ⊗ \|1, 0,m− 1〉 + q(1− q2m)(qmz − ql) (ql+m − z)(ql+m − q2z) \|1, 0, l − 1〉 ⊗ \|0, 1,m− 1〉 + (qmz − ql)(qm+2z − ql) (ql+m − z)(ql+m − q2z) \|1, 1, l − 2〉 ⊗ \|0, 0,m〉, R(z)(\|1, 1, l − 2〉 ⊗ \|0, 1,m− 1〉) = (1− ql+mz)(1− q2m−2) (ql+m − z)(ql+m−2 − z) \|0, 1, l − 1〉 ⊗ \|1, 1,m− 2〉 + (1− ql+mz)(qm−1z − ql−1) (ql+m − z)(ql+m−2 − z) \|1, 1, l − 2〉 ⊗ \|0, 1,m− 1〉, R(z)(\|1, 1, l − 2〉 ⊗ \|1, 0,m− 1〉) = (1− ql+mz)(1− q2m−2) (ql+m − z)(ql+m−2 − z) \|1, 0, l − 1〉 ⊗ \|1, 1,m− 2〉 + (1− ql+mz)(qm−1z − ql−1) (ql+m − z)(ql+m−2 − z) \|1, 1, l − 2〉 ⊗ \|1, 0,m− 1〉, R(z)(\|1, 1, l − 2〉 ⊗ \|1, 1,m− 2〉) = (1− ql+mz)(1− ql+m−2z) (ql+m − z)(ql+m−2 − z) \|1, 1, l − 2〉 ⊗ \|1, 1,m− 2〉, R(z)(\|1, 0, l − 1〉 ⊗ \|0, 0,m〉) = 1− q2m z − ql+m \|0, 0, l〉 ⊗ \|1, 0,m− 1〉 + qmz − ql z − ql+m \|1, 0, l − 1〉 ⊗ \|0, 0,m〉, Integrable Structure of Multispecies Zero Range Process 27 R(z)(\|1, 0, l − 1〉 ⊗ \|0, 1,m− 1〉) = (q2m − q2)(qm − qlz) (ql+m − z)(ql+m − q2z) \|0, 0, l〉 ⊗ \|1, 1,m− 2〉 + (q2 − 1)ql+m + (q2 − q2+2l − q2+2m + q2l+2m)z (ql+m − z)(ql+m − q2z) × \|0, 1, l − 1〉 ⊗ \|1, 0,m− 1〉 + q(qm − qlz)(ql − qmz) (ql+m − z)(ql+m − q2z) \|1, 0, l − 1〉 ⊗ \|0, 1,m− 1〉 + (q2l − q2)(ql − qmz)z (ql+m − z)(ql+m − q2z) \|1, 1, l − 2〉 ⊗ \|0, 0,m〉, R(z)(\|1, 0, l − 1〉 ⊗ \|1, 0,m− 1〉) = 1− ql+mz z − ql+m \|1, 0, l − 1〉 ⊗ \|1, 0,m− 1〉, R(z)(\|1, 0, l − 1〉 ⊗ \|1, 1,m− 2〉) = q(1− ql+mz)(qlz − qm) (ql+m − z)(ql+m − q2z) \|1, 0, l − 1〉 ⊗ \|1, 1,m− 2〉 + q2z(1− ql+mz)(1− q2l−2) (ql+m − z)(ql+m − q2z) \|1, 1, l − 2〉 ⊗ \|1, 0,m− 1〉, R(z)(\|0, 1, l − 1〉 ⊗ \|0, 0,m〉) = 1− q2m z − ql+m \|0, 0, l〉 ⊗ \|0, 1,m− 1〉 + qmz − ql z − ql+m \|0, 1, l − 1〉 ⊗ \|0, 0,m〉, R(z)(\|0, 1, l − 1〉 ⊗ \|0, 1,m− 1〉) = 1− ql+mz z − ql+m \|0, 1, l − 1〉 ⊗ \|0, 1,m− 1〉, R(z)(\|0, 1, l − 1〉 ⊗ \|1, 0,m− 1〉) = q(qlz − qm)(1− q2m−2) (ql+m − z)(ql+m − q2z) \|0, 0, l〉 ⊗ \|1, 1,m− 2〉 + q(qlz − qm)(qmz − ql) (ql+m − z)(ql+m − q2z) \|0, 1, l − 1〉 ⊗ \|1, 0,m− 1〉 + ql+m(1− q2)z2 + (q2 + q2l+2m − q2l − q2m)z (ql+m − z)(ql+m − q2z) × \|1, 0, l − 1〉 ⊗ \|0, 1,m− 1〉 + qz(1− q2l−2)(qmz − ql) (ql+m − z)(ql+m − q2z) \|1, 1, l − 2〉 ⊗ \|0, 0,m〉, R(z)(\|0, 1, l − 1〉 ⊗ \|1, 1,m− 2〉) = (1− ql+mz)(ql−1z − qm−1) (z − ql+m)(z − ql+m−2) \|0, 1, l − 1〉 ⊗ \|1, 1,m− 2〉 + (1− ql+mz)(1− q2l−2)z (z − ql+m)(z − ql+m−2) \|1, 1, l − 2〉 ⊗ \|0, 1,m− 1〉. 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Integrable Structure of Multispecies Zero Range Process

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