aa r X i v : . [ m a t h - ph ] F e b Stochastic symplectic ice
Chenyang Zhong ∗ January 31, 2021
Abstract
In this paper, we construct solvable ice models (six-vertex models) with stochastic weights and U-turnright boundary, which we term “stochastic symplectic ice”. The models consist of alternating rows of twotypes of vertices. The probabilistic interpretation of the models offers novel interacting particle systemswhere particles alternately jump to the right and then to the left. Two colored versions of the modeland related stochastic dynamics are also introduced. Using the Yang-Baxter equations, we establishfunctional equations and recursive relations for the partition functions of these models. In particular,the recursive relations satisfied by the partition function of one of the colored models are closely relatedto Demazure-Lusztig operators of type C.
Since the pioneering investigation by Baxter ([1],[2]), exactly solvable lattice models have found applicationsto various fields of mathematics and mathematical physics. “Exactly solvable” means that the Yang-Baxterequation, or “star-triangle relation”, is satisfied by the system. We refer the reader to [28],[10],[23],[24] forapplications of exactly solvable models to many algebraic combinatorics problems.Recently, there has been a series of work (see, for example, [13],[11],[6],[9],[7],[8]) relating representationtheory (for example, Tokuyama-type formulas and non-archimedean Whittaker functions) to exactly solvablelattice models. These models are for Cartan type A, and are based on the Yang-Baxter equation for freeFermionic six-vertex models ([13]).For Cartan type C, a parallel line of work has been initiated by Hamel and King ([20],[21]) and Ivanov([22]). These works showed that the partition function of certain ice model (six-vertex model) equals the prod-uct of a deformation of Weyl’s denominator and an irreducible character of the symplectic group Sp(2 n, C ).Ivanov’s approach used the Yang-Baxter equation developed in [13]. His lattice model consists of alternatingrows of two types of ice (called Γ ice and ∆ ice) and has U-turn boundary on the right end. This U-turn typemodel was also applied to Whittaker functions on the metaplectic double cover of Sp(2 n, F ) where F is anon-archimedean local field ([12]). Later work ([19]) extended the results to metaplectic ice for Cartan typeC. In [15], new deformations of Weyl’s character formula for Cartan type B,C and D, and a character formulaof Proctor for type BC, were obtained using the Yang-Baxter equation. A different approach based on adiscrete time evolution operator on one-dimensional Fermionic Fock space is in [16]. Further developments,including dual wave function of the symplectic ice and generalizations of the ice models in [22] and [12], arein [25],[26].Another sequence of recent work, which comes from the subject of “integrable probability”, relatesstochastic systems such as the asymmetric simple exclusion processes (ASEP) and the KPZ equation, tosolvable lattice models called the stochastic higher spin six-vertex models (see, for example, [3],[18]). Exactlysolvable lattice models provide a powerful tool for analyzing probabilistic properties of these stochasticsystems (for example, for proving Tracy-Widom type fluctuation results). The reader is also referred to [4]for a useful tutorial. The works on stochastic vertex models so far have been mainly restricted to modelsthat are related to Cartan type A. ∗ Department of Statistics, Stanford University U q ( [ sl n +1 ), and degen-erate to multi-species versions of interacting particle systems (such as multi-species ASEP). In [5], recursiverelations for the partition functions of these colored models were derived using the Yang-Baxter equation,and were related to Demazure-Lusztig operators (of type A).In this paper, we also construct two colored versions of the stochastic symplectic ice model. The coloredmodels are also stochastic, and can be interpreted as stochastic dynamics of interacting particles with colors.In one of the colored models (which we introduce and study in Section 3), each particle carries a “signedcolor”, whose sign changes to the opposite when reflected from the U-turn boundary. This seems to be a novelfeature compared to previous colored stochastic vertex models. When specifying the boundary conditions,two signed permutations from the hyperoctahedral group (which is the Weyl group for type C) are involved.Previous works have mainly been focusing on boundary conditions specified by the symmetric group (whichis the Weyl group for type A).The colored models are also solvable, in that three sets of the Yang-Baxter equations are satisfied bythe models. These equations, when combined with a further relation, the reflection equation, allow us toobtain recursive relations for the partition functions. The recursive relations for one of the colored modelsare further related to Demazure-Lusztig operators of type C.A few days before this paper is posted, a preprint by Buciumas and Scrimshaw ([17]) appeared on thearXiv. Their work constructed colored lattice models with partition functions representing symplectic andodd orthogonal Demazure characters and atoms. The work in this paper was done independently of andconcurrently with their work. We also note that the colored models in this paper are quite different fromtheirs: the colored models in this paper are six-vertex models, while their models are five-vertex models(in that the b patterns for both Γ ice and ∆ ice have Boltzmann weight 0 in their paper); the Boltzmannweights in this paper are also quite different from theirs; the recursive relations in this paper are related toDemazure-Lusztig operators of type C, while theirs are related to Demazure atoms and characters.Section 2 of the paper introduces the two types of the stochastic symplectic ice model. The Yang-Baxterequations, the caduceus relation and the fish relation are presented there. These relations are used to derivethe functional equations for the partition functions. Section 3 introduces one of the colored models for thestochastic symplectic ice. The Yang-Baxter equations and the reflection equation are also introduced in thissection. Using these relations, the recursive relations for the partition functions are obtained. The recursiverelations are further related to Demazure-Lusztig operators of type C. Finally we introduce and study theother colored model in Section 4. 2 a b b c c + + ++ z i − − −− z i + − + − z i − + − + z i − + + − z i + − − + z i z i qz i − qz i − z i Figure 1: Boltzmann weights for stochastic Γ vertex with spectral parameter z i The author wishes to thank Daniel Bump for his encouragement and many helpful conversations.
In this section, we introduce and study two classes of stochastic symplectic ice. They are termed “reflectingstochastic symplectic ice” and “absorbing-and-emitting stochastic symplectic ice”. Section 2.1 introducesthe models and related Boltzmann weights. Section 2.2 gives the Boltzmann weights for the R-matricesand shows the Yang-Baxter equation. An additional relation, the “caduceus relation”, is shown in Section2.3. By combining the Yang-Baxter equation with the caduceus relation and a further relation, the “fishrelation”, we establish functional equations satisfied by the partition functions in Section 2.4.
First we introduce some notations. By “ice model”, we mean a planar lattice where every edge is assigned a+ or − spin. To each vertex in the lattice, we assign a Boltzmann weight, which is a number that depends onthe type of the vertex (there are two types of vertices for stochastic symplectic ice, see the next paragraph)and the + or − spins assigned to the four adjacent edges. A configuration/state means a labeling of the edgesof the graph by + or − spins, and the Boltzmann weight of a configuration is the product of the Boltzmannweights of all the vertices for the configuration. An admissible state is a state where the assignment ofspins to the edges adjacent to each vertex is one of the allowed assignments for that vertex (the allowedassignments are listed in tables later in the paper). The partition function of the ice model is the sum ofBoltzmann weights for all admissible configurations.In the stochastic symplectic ice model, two types of vertices are involved. They are termed “stochasticΓ vertex” and “stochastic ∆ vertex” in this paper, in analogy to the Γ ice and the ∆ ice used in Ivanov’ssymplectic ice model (see [22]). The model depends on n + 1 parameters z , · · · , z n , q , where z , · · · , z n arecalled “spectral parameters” and q is called “deformation parameter”. Throughout the paper, we also define z ′ i = q + 1 − z i , for every 1 ≤ i ≤ n .The Boltzmann weights for the stochastic Γ vertex and the stochastic ∆ vertex (with spectral parameter z i ) are listed in Figures 1-2. Throughout the paper, an assignment of spins to the adjacent edges of a vertexthat is not listed in the corresponding table has Boltzmann weight 0.Now we introduce the stochastic symplectic ice model. We consider a rectangular lattice with 2 n rowsand L columns. The rows are numbered 1 , , · · · , n from bottom to top, and the columns are numbered1 , , · · · , L from right to left. Every odd-numbered row is a row of stochastic ∆ vertices, and every even-numbered row is a row of stochastic Γ vertices. The spectral parameter for the i th row of stochastic Γvertices and the i th row of stochastic ∆ vertices is given by z i .3 a b b d d + + ++ z i − − −− z i + − + − z i − + − + z i − − ++ z i + + −− z i z ′ i q z ′ i − z ′ i − q z ′ i Figure 2: Boltzmann weights for stochastic ∆ vertex with spectral parameter z i , where z ′ i = q + 1 − z i The model also depends on a partition λ = ( λ , · · · , λ n ′ ) ∈ Z n ′ , where λ ≥ · · · ≥ λ n ′ and n ′ ∈ N + . Weassume that L ≥ λ + n ′ . The assignment of spins to boundary edges of the rectangular lattice is given asfollows: on the left column, we assign − to each row of stochastic Γ vertex, and + to each row of stochastic∆ vertex; on the top, we assign + to each boundary edge; on the bottom, we assign − to each column labeled λ i + n ′ + 1 − i , for 1 ≤ i ≤ n ′ ; on the right, the i th row of stochastic Γ vertices and the i th row of stochastic∆ vertices is connected by a “cap”.For example, when n = 2, λ = (2 , L = 4, the model configuration is shown in Figure 3.+ − + − z z z z + + + +4 3 2 1column − + − +Figure 3: Model configuration when n = 2 , λ = (2 , , L = 4We now discuss the Boltzmann weights for the caps. There are two choices of Boltzmann weights forthe caps, which lead to two types of stochastic symplectic ice: “reflecting stochastic symplectic ice” and“absorbing-and-emitting stochastic symplectic ice”. The Boltzmann weights of the caps for the two modelsare listed in Figures 4-5, respectively. For reflecting symplectic ice, we always assume that n ′ = n whentaking the partition λ for boundary conditions (by particle conservation).Cap ++ −− Boltzmann weight 1 1Figure 4: Boltzmann weights for caps: reflecting stochastic symplectic iceThroughout the paper, we denote by z = ( z , z , · · · , z n ) the vector of the n spectral parameters. Wealso denote by S n,L,λ,z the collection of admissible configurations of the reflecting symplectic ice with 2 n − + + − Boltzmann weight 1 1Figure 5: Boltzmann weights for caps: absorbing-and-emitting stochastic symplectic icerows, L columns, spectral parameters z , · · · , z n and bottom boundary condition given by λ . We alsolet Z ( S n,L,λ,z ) be the corresponding partition function. The collection of admissible configurations andthe partition function of the absorbing-and-emitting stochastic symplectic ice are denoted by T n,L,λ,z and Z ( T n,L,λ,z ), respectively.We note that the Boltzmann weights of our model are stochastic. For stochastic Γ vertex, we view theleft and top edges adjacent to the vertex as input, and the other two as output; for stochastic ∆ vertex, weview the right and top edges adjacent to the vertex as input, and the other two as output; for caps, we viewthe top spin as input and the bottom spin as output. It can be seen that the possible Boltzmann weightsfor a given vertex (either the stochastic Γ vertex, the stochastic ∆ vertex or the cap) with given input sumup to 1. Moreover, if z satisfies the conditionmax { , q + 1 } ≤ z i ≤ min { q , } , for every 1 ≤ i ≤ n, (2.1)then all possible Boltzmann weights are non-negative. Therefore, when the condition (2.1) is satisfied, theBoltzmann weight of a given vertex can be interpreted as the probability of obtaining the output given theinput at that vertex.We also note that if the condition (2.1) is satisfied by z , then both types of stochastic symplectic ice canbe interpreted as an interacting particle system. We put x − y coordinates on the rectangular lattice (see,for example, Figure 3) such that the x coordinate of the i th row is i and the y coordinate of the j th columnis j . For each t = 0 , , · · · , n , we consider the set of vertical edges of the lattice that have a non-emptyintersection with the line x = 2 n − t + and carry a − spin. The positions of the particles at time t are justthe y coordinates of these vertical edges. Therefore, an admissible state of the stochastic symplectic ice givesa possible evolution of the particles, and the Boltzmann weight for that state represents the probability ofthe particular evolution.Now we describe the stochastic dynamics of the particles. For t = 0 , , · · · , n −
1, if t is even, theparticles attempt to jump to the right; if t is odd, the particles attempt to jump to the left.The detailed rule is as follows. When t is even, the particles are ordered from left to right. There is a newparticle entering from the left boundary (we call it 0th particle), which jumps to the right with geometricjump size (with parameter qz n − t ) unless it hits the 1st particle; if the particle hits the 1st particle, thenit stops to move further. Starting from l = 1, if the l th particle wasn’t hit by any particle on its left, weflip a coin with head probability z n − t to determine whether it will stay at its current position or not; if thecoin comes up tail, then the particle jumps to the right with geometric jump size (with parameter qz n − t )unless it hits the ( l + 1)th particle; if the particle hits the ( l + 1)th particle, then it stops to move further.If the l th particle was hit by the ( l − l + 1)th particle begins to move. If the rightmost particle moves beyond the first column (meaningthat it hits the cap), for reflecting symplectic ice it is reflected by the cap (so it will start to move leftwardfrom the first column at time t + 1), while for absorbing-and-emitting symplectic ice it is absorbed. Forabsorbing-and-emitting symplectic ice, if no particle hits the cap, a new particle will be emitted from thefirst column at time t + 1.When t is odd, the particles are ordered from right to left. If there is a particle reflected/emitted fromthe cap (we call it 0th particle), it jumps to the left with geometric jump size (with parameter q z ′ n − t − )5nless it hits the 1st particle; if the particle hits the 1st particle, then it stops to move further. Starting from l = 1, if the l th particle wasn’t hit by any particle on its right, we flip a coin with head probability z ′ n − t − to determine whether it will stay at its current position or not; if the coin comes up tail, then the particlejumps to the left with geometric jump size (with parameter q z ′ n − t − ) unless it hits the ( l + 1)th particle;if the particle hits the ( l + 1)th particle, then it stops to move further. If the l th particle was hit by the( l − l + 1)th particle beginsto move.Under this probabilistic interpretation, the partition functions Z ( S n,L,λ,z ) and Z ( T n,L,λ,z ) represent theprobability that the particle configuration at time t = 2 n corresponds to the partition λ (meaning that the i th particle, ordered from left to right, has coordinate λ i + n ′ + 1 − i for 1 ≤ i ≤ n ′ ) and no particle has evermoved left of the L th column. The Yang-Baxter equation is a powerful tool for studying solvable lattice models. It involves two ordinaryvertices (for our model, the stochastic Γ vertex or the stochastic ∆ vertex) and one additional rotated vertexcalled the R-vertex (also called the “R-matrix”, due to connections to quantum group theory).The stochastic symplectic ice, as introduced in Section 2.1, is a solvable lattice model, in that we canfind four types of R-matrices such that four sets of Yang-Baxter equations are satisfied by the model. Inthis section, we introduce the R-matrices for the stochastic symplectic ice, and show that the R-matricestogether with the ordinary vertices (stochastic Γ and ∆ vertex) satisfy the Yang-Baxter equations, in theform of Theorem 2.1 below.The four types of R-matrices are termed “stochastic Γ − Γ vertex”, “stochastic Γ − ∆ vertex”, “stochastic∆ − ∆ vertex” and “stochastic ∆ − Γ vertex”, in analogy to the terms used in [22] for the symplectic ice.The Boltzmann weights for these R-matrices are given in Figures 6-9.++ +++ R z i ,z j −− −− R z i ,z j + − + − R z i ,z j − + − + R z i ,z j − + + − R z i ,z j + − − + R z i ,z j z i − z j − ( q +1) z j + qz i z j q ( z i − z j )1 − ( q +1) z j + qz i z j (1 − qz i )(1 − z j )1 − ( q +1) z j + qz i z j (1 − z i )(1 − qz j )1 − ( q +1) z j + qz i z j Figure 6: Boltzmann weights for stochastic Γ − Γ vertex with spectral parameters z i and z j ++ +++ R z i ,z j −− −− R z i ,z j + − + − R z i ,z j − + − + R z i ,z j −− ++ R z i ,z j ++ −− R z i ,z j z ′ i + qz j − ( q +1) z ′ i z j − z ′ i z j q − z ′ i + z j − (1+ q − ) z ′ i z j − z ′ i z j (1 − z ′ i )(1 − qz j )1 − z ′ i z j (1 − q − z ′ i )(1 − z j )1 − z ′ i z j Figure 7: Boltzmann weights for stochastic ∆ − Γ vertex with spectral parameters z i and z j The following theorem gives the four sets of Yang-Baxter equations for the stochastic symplectic ice.6+ +++ R z i ,z j −− −− R z i ,z j + − + − R z i ,z j − + − + R z i ,z j − + + − R z i ,z j + − − + R z i ,z j z ′ j − z ′ i q − ( q +1) z ′ i + z ′ i z ′ j q ( z ′ j − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ i )( q − z ′ j ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ j )( q − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j Figure 8: Boltzmann weights for stochastic ∆ − ∆ vertex with spectral parameters z i and z j ++ +++ R z i ,z j −− −− R z i ,z j + − + − R z i ,z j − + − + R z i ,z j −− ++ R z i ,z j ++ −− R z i ,z j qz i + z ′ j − (1+ q ) z i z ′ j − qz i + z ′ j − (1+ q ) q ( z i z ′ j −
1) (1 − qz i )(1 − z ′ j ) z i z ′ j − − z i )( q − z ′ j ) q ( z i z ′ j − Figure 9: Boltzmann weights for stochastic Γ − ∆ vertex with spectral parameters z i and z j Theorem 2.1.
For any
X, Y ∈ { Γ , ∆ } the following holds. Assume that S is stochastic X vertex withspectral parameter z i , T is stochastic Y vertex with spectral parameter z j , and R is stochastic X − Y vertexwith spectral parameters z i , z j . Then the partition functions of the following two configurations are equal forany fixed combination of spins a, b, c, d, e, f . ab c defg hi STR ab c def jk lTS R (2.2) Proof.
There are in total 2 = 64 possible combinations of the boundary spins a, b, c, d, e, f . These identitiesare checked using a SAGE program. In addition to the Yang-Baxter equation, the stochastic symplectic ice also satisfies a further relation calledthe “caduceus relation”, which plays an important role in deriving functional equations for the partitionfunctions in Section 2.4. Namely, we have the following
Theorem 2.2.
Assume that A is stochastic Γ − Γ vertex, B is stochastic ∆ − ∆ vertex, C is stochastic ∆ − Γ vertex, and D is stochastic Γ − ∆ vertex. Also assume that the spectral parameters of the four vertices A, B, C, D are all z i , z j . Denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) the partition function of the following configuration ith fixed combination of spins ǫ , ǫ , ǫ , ǫ . I ( ǫ , ǫ , ǫ , ǫ ) = D CBAǫ ǫ ǫ ǫ (2.3) Also denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) the partition function of the following configuration with fixed combinationof spins ǫ , ǫ , ǫ , ǫ . I ( ǫ , ǫ , ǫ , ǫ ) = ǫ ǫ ǫ ǫ (2.4) Then for any fixed combination of spins ǫ , ǫ , ǫ , ǫ , and for both choices of cap weights given in Figures4-5, we have Z ( I ( ǫ , ǫ , ǫ , ǫ )) = ( qz i z j − − ( q + 1)( z i + z j ) + ( q + q + 1) z i z j ) q ( z i + z j − ( q + 1) z i z j ) Z ( I ( ǫ , ǫ , ǫ , ǫ )) . (2.5) Proof.
The are in total 2 = 16 possible combinations of the boundary spins ǫ , ǫ , ǫ , ǫ . The identities arechecked using a SAGE program. In this section, we derive functional equations satisfied by the partition functions. The main result is thefollowing
Theorem 2.3.
Let D ( n, L, z ) = n Y i =1 z Li n Y i =1 (1 − ( q + 1) z i + qz i z ′− i ) (2.6) and D ( n, L, z ) = n Y i =1 z Li . (2.7) Then Z ( S n,L,λ,z ) D ( n,L,z ) and Z ( T n,L,λ,z ) D ( n,L,z ) are invariant under any permutation of z , · · · , z n and any interchange z i ↔ z ′− i . Theorem 2.3 follows from Propositions 2.4-2.5 below. Proposition 2.4 gives a functional equation when z , · · · , z n are permuted, and Proposition 2.5 gives another functional equation under the interchange z n ↔ z ′ n . Note that z i + z ′ i = q + 1 for every 1 ≤ i ≤ n . Proposition 2.4.
The partition functions of the two types of stochastic symplectic ice, namely, Z ( S n,L,λ,z ) and Z ( T n,L,λ,z ) , are both invariant under any permutation of z , · · · , z n . roposition 2.5. Let s n z := ( z , · · · , z n − , z ′ n ) . Then we have Z ( S n,L,λ,s n z ) = ( 1 z n z ′ n ) L − ( q + 1) z ′− n + qz n z ′− n − ( q + 1) z n + qz n z ′− n Z ( S n,L,λ,z ) , (2.8) Z ( T n,L,λ,s n z ) = ( 1 z n z ′ n ) L Z ( T n,L,λ,z ) . (2.9)The rest of this section is devoted to the proof of Propositions 2.4-2.5. The proof of Proposition 2.4 isbased on the Yang-Baxter equation and the caduceus relation. The proof of Proposition 2.5 is based on theYang-Baxter equation and another relation called the “fish relation” (see Proposition 2.7 below). Based on Theorem 2.1 and Theorem 2.2, we give the proof of Proposition 2.4 as follows.
Proof of Proposition 2.4.
We note that the symmetric group S n is generated by adjacent transpositions( i, i + 1) for 1 ≤ i ≤ n −
1. Therefore it suffices to verify the invariance of the partition functions under thetransposition of z i and z i +1 . We show the details below for Z ( S n,L,λ,z ). The argument for Z ( T n,L,λ,z ) isessentially the same.We attach a braid to the left boundary of the rows 2 i − , i, i + 1 , i + 2 of S n,L,λ,z , and obtain thefollowing D CBA + − + − z i z i z i +1 z i +1 ∆Γ∆Γ (2.10)where we have omitted the other rows of S n,L,λ,z . We denote by Z ( J ) the partition function of this new icemodel.Note that the only admissible configuration of the braid is given as follows: D CBA + − + − z i z i z i +1 z i +1 ∆Γ∆Γ −− ++ − + − + (2.11)Therefore Z ( J ) is the product of the partition function of the braid and Z ( S n,L,λ,z ). Let L ( z, q, i ) = ( qz i z i +1 − − ( q + 1)( z i + z i +1 ) + ( q + q + 1) z i z i +1 ) q ( z i + z i +1 − ( q + 1) z i z i +1 ) . (2.12)By computation, we obtain that Z ( J ) = L ( z, q, i ) Z ( S n,L,λ,z ) . (2.13)9ow using the four sets of Yang-Baxter equations (Theorem 2.1), we can move the four vertices (in theorder of C, A, B, D ) of the braid to the right without changing the partition function. Namely, if we denoteby Z ( J ) the partition function of the following D CBA + − + − z i +1 z i +1 z i z i ∆Γ∆Γ (2.14)then we have that Z ( J ) = Z ( J ) (2.15)Let J ( ǫ , ǫ , ǫ , ǫ ) be given as follows, and recall the definition of I ( ǫ , ǫ , ǫ , ǫ ) and I ( ǫ , ǫ , ǫ , ǫ )from the statement of Theorem 2.2 (taking the spectral parameters to be z i +1 , z i ). J ( ǫ , ǫ , ǫ , ǫ ) = + − + − z i +1 z i +1 z i z i ∆Γ∆Γ ǫ ǫ ǫ ǫ (2.16)Now note that by Theorem 2.2, Z ( J ) = X ǫ ,ǫ ,ǫ ,ǫ ∈{− , + } Z ( I ( ǫ , ǫ , ǫ , ǫ )) Z ( J ( ǫ , ǫ , ǫ , ǫ ))= L ( z, q, i ) X ǫ ,ǫ ,ǫ ,ǫ ∈{− , + } Z ( I ( ǫ , ǫ , ǫ , ǫ )) Z ( J ( ǫ , ǫ , ǫ , ǫ ))= L ( z, q, i ) Z ( S n,L,λ,s i z ) , (2.17)where s i z is the vector obtained by interchanging z i , z i +1 from z .By combining (2.13),(2.15),(2.17) we obtain that Z ( S n,L,λ,z ) = Z ( S n,L,λ,s i z ) , (2.18)which finishes the proof. Before the proof of Proposition 2.5, we make the following observation. As all the boundary edges on thetop carry the + spin, we conclude that only the three states in Figure 10 are involved in the 2 n th row. Nowwe simultaneously change the sign of the spins in the 2 n th row (interchanging − and + spins), change theBoltzmann weights of the vertices in the 2 n th row to those in Figure 11, and change the Boltzmann weightsfor the cap connecting the last two rows to those in Figure 12 or 13 (depending on the type of the stochastic10ymplectic ice). For each admissible state, the Boltzmann weight of each vertex in the 2 n th row is nowscaled by a factor of qz n . Therefore the partition functions of the new system, denoted by Z ( S ′ n,L,λ,z ) and Z ( T ′ n,L,λ,z ) respectively, satisfy the following Z ( S ′ n,L,λ,z ) = 1( qz n ) L Z ( S n,L,λ,z ) , (2.19) Z ( T ′ n,L,λ,z ) = 1( qz n ) L Z ( T n,L,λ,z ) . (2.20)+ + ++ z n − + − + z n − + + − z n qz n − qz n Figure 10: Weights involved in the 2 n th row a a b b d d + + ++ z n − − −− z n + − + − z n − + − + z n − − ++ z n + + −− z n z n qz n z n − qz n − n th rowNew cap − + + − Boltzmann weight 1 1Figure 12: New Boltzmann weights for the cap connecting the last two rows: reflecting stochastic symplecticice The following lemma gives a new set of Yang-Baxter equations, which will be used in the proof ofProposition 2.5.
Lemma 2.6.
Assume that t , t ∈ C . Also assume that the Boltzmann weights of S are given by Figure 14,the Boltzmann weights of T are given by Figure 15, and the Boltzmann weights of R are given by Figure 16.Then the partition functions of the following two configurations are equal for any fixed combination of spins a, b, c, d, e, f . −− Boltzmann weight 1 1Figure 13: New Boltzmann weights for the cap connecting the last two rows: absorbing-and-emitting stochas-tic symplectic ice a a b b d d + + ++ − − −− + − + − − + − + − − ++ + + −− qt t qt − t − S : Lemma 2.6 ab c defg hi STR ab c def jk lTS R (2.21) Proof.
There are 2 = 64 possible combinations of boundary spins. We have checked the identities using aSAGE program.Now consider the R-matrix with Boltzmann weights given by Figure 17. It is obtained by taking t = qz n and t = q z ′ n in the Boltzmann weights from Figure 16. The following theorem gives the “fish relation”satisfied by the new R-matrix and the new cap. Proposition 2.7.
Suppose the Boltzmann weights of R in the following is given by Figure 17. Denote by Z ( I ( ǫ , ǫ )) the partition function of the following system. I ( ǫ , ǫ ) = ǫ ǫ R (2.22) Also denote by Z ( I ( ǫ , ǫ )) the partition function of the following system. I ( ǫ , ǫ ) = ǫ ǫ (2.23)12 a b b d d + + ++ − − −− + − + − − + − + − − ++ + + −− qt t − qt − t Figure 15: Boltzmann weights for T : Lemma 2.6 a a b b c c ++ +++ −− −− + − + − − + − + − + + − + − − +1 1 t − t − ( q +1) t + qt t q ( t − t )1 − ( q +1) t + qt t − (1 − t )(1 − qt )1 − ( q +1) t + qt t − (1 − t )(1 − qt )1 − ( q +1) t + qt t Figure 16: Boltzmann weights for R : Lemma 2.6 Then for reflecting stochastic symplectic ice (i.e. the Boltzmann weights for the new cap are given by Figure12), we have Z ( I ( ǫ , ǫ )) = − − ( q + 1) z n + qz n z ′− n − ( q + 1) z ′− n + qz n z ′− n Z ( I ( ǫ , ǫ )); (2.24) for absorbing-and-emitting stochastic symplectic ice (i.e. the Boltzmann weights for the new cap are givenby Figure 13), we have Z ( I ( ǫ , ǫ )) = Z ( I ( ǫ , ǫ )) . (2.25) Proof.
We denote by a , a , b , b , c , c the Boltzmann weights for the R-matrix.First consider reflecting stochastic symplectic ice. In this case Z ( I (+ , +)) = Z ( I ( − , − )) = Z ( I (+ , +)) = Z ( I ( − , − )) = 0. Moreover, Z ( I (+ , − )) = c + b = − − ( q + 1) z n + qz n z ′− n − ( q + 1) z ′− n + qz n z ′− n Z ( I (+ , − )) , (2.26) Z ( I ( − , +)) = c + b = − − ( q + 1) z n + qz n z ′− n − ( q + 1) z ′− n + qz n z ′− n Z ( I ( − , +)) . (2.27)Now consider absorbing-and-emitting stochastic symplectic ice. In this case Z ( I (+ , − )) = Z ( I ( − , +)) = Z ( I (+ , − )) = Z ( I ( − , +)) = 0. Moreover, Z ( I (+ , +)) = a = Z ( I (+ , +)) , (2.28) Z ( I ( − , − )) = a = Z ( I ( − , − )) . (2.29)We finish the proof of Proposition 2.5 as follows. 13 a b b c c ++ +++ −− −− + − + − − + − + − + + − + − − +1 1 z n z ′ n − qz n + z ′ n − ( q +1) q ( z n z ′ n − qz n + z ′ n − ( q +1) − ( z n − q − z ′ n ) qz n + z ′ n − ( q +1) − ( qz n − − z ′ n ) qz n + z ′ n − ( q +1) Figure 17: The R-matrix used in the proof of Proposition 2.5
Proof of Proposition 2.5.
For ease of notations, we denote by V ( a , a , b , b , c , c , d , d ) a vertex withBoltzmann weights given by a , a , b , b , c , c , d , d .Now we attach the R-matrix given by Figure 17 to the left boundary of the last two rows of the changedsystem: ++ R V (1 , , z n , qz n , , , z n − , qz n − V (1 , , z ′ n , q z ′ n , , , − z ′ n , − q z ′ n )+ (2.30)Note that the only admissible configuration of the R-matrix is given by++ R V (1 , , z n , qz n , , , z n − , qz n − V (1 , , z ′ n , q z ′ n , , , − z ′ n , − q z ′ n )+ ++ (2.31)Therefore, the partition function of the above system is equal to Z ( S ′ n,L,λ,z ) or Z ( T ′ n,L,λ,z ) according to thetype of the stochastic symplectic ice.By Lemma 2.6, the R-matrix can be pushed to the right without changing the partition function. Thatis, the partition function of the above system is equal to the partition function of the following++ RV (1 , , z n , qz n , , , z n − , qz n − V (1 , , z ′ n , q z ′ n , , , − z ′ n , − q z ′ n )+ (2.32)Consider the reflecting stochastic symplectic ice. By Proposition 2.7, the above partition function isequal to − − ( q +1) z n + qz n z ′− n − ( q +1) z ′− n + qz n z ′− n times the partition function of the following system (denoted by Z )++ V (1 , , z n , qz n , , , z n − , qz n − V (1 , , z ′ n , q z ′ n , , , − z ′ n , − q z ′ n )+ (2.33)We note again that the top boundary edges of the system all carry + spin. We change the sign of thespins of the 2 n th row again (interchanging + and − ), and also change the Boltzmann weights of the 2 n th14ow as in the following configuration. The Boltzmann weights for the cap connecting the last two rows arechanged back to the original one given in Figure 4. We denote the partition function of the following systemby Z . + − V (1 , , z n , qz n , , , z n − , qz n − V (1 , , z ′ n , qz ′ n , qz ′ n − , z ′ n − , ,
0) (2.34)Similar to the previous argument, we conclude that Z = ( qz ′ n ) L Z . (2.35)Now note that the total number of c , c , d , d patterns in the last two rows is an odd number (as canbe seen by interpreting − spins as paths and considering all possibilities). Hence Z is equal to − Z ( S n,L,λ,s n z ).+ − V (1 , , z n , qz n , , , − z n , − qz n ) V (1 , , z ′ n , qz ′ n , − qz ′ n , − z ′ n , ,
0) (2.36)Therefore we conclude that Z ( S n,L,λ,s n z ) = ( 1 z n z ′ n ) L − ( q + 1) z ′− n + qz n z ′− n − ( q + 1) z n + qz n z ′− n Z ( S n,L,λ,z ) . (2.37)The conclusion for Z ( T n,L,λ,s n z ) can be obtained similarly, noting that the number of c , c , d , d patternsin the last two rows is an even number for this case. In this section, we introduce a colored version of the stochastic symplectic ice model. For each edge of therectangular lattice, instead of assigning either a + or − spin, we now associate either + or one of 2 n colorsto it. The 2 n colors are labeled by [ ± n ] = { n, · · · , , , · · · , n } .The colored model is closely related to Cartan type C: part of the boundary conditions are specifiedby two elements of the hyperoctahedral group, the Weyl group of type C; the recursive relations for thepartition function, upon a change of variables, are related to Demazure-Lusztig operators of type C.We start by introducing the colored model in Section 3.1. Then we introduce the R-matrix and prove theYang-Baxter equation in Section 3.2. In Section 3.3, we compute the partition function when σ ( i ) = τ ( i )for every 1 ≤ i ≤ n . Then we present a new relation, the reflection equation, in Section 3.4. By combingthe Yang-Baxter equation and the reflection equation, we derive the recursive relations for the partitionfunction in Section 3.5. The recursive relations are further related to Demazure-Lusztig operators of type Cin Section 3.6.We briefly introduce the hyperoctahedral group, denoted by B n , here. The hyperoctahedral group hasthe following presentation B n = h s , · · · , s n | s i = 1 , ≤ i ≤ n ; ( s i s i +1 ) = 1 , ≤ i ≤ n − s n − s n ) = 1; ( s i s j ) = 1 , ≤ i < j ≤ n, | i − j | > i . B n is the Weyl group for the root system of type C n . Elements of B n can be viewed as permu-tations σ of [ ± n ] such that σ ( − i ) = − σ ( i ) for every 1 ≤ i ≤ n .We make the convention that elements of B n are multiplied from right to left, and that i = i for each1 ≤ i ≤ n . Thus for each i ∈ [ ± n ] and σ, τ ∈ B n , we have στ ( i ) = σ ( τ ( i )). Note that s i is the transposition( i, i + 1) for each 1 ≤ i ≤ n −
1, and that s n ( i ) = i for 1 ≤ i ≤ n − s n ( n ) = n . We introduce the colored version of the stochastic symplectic ice in this section. The main difference fromthe uncolored model is that now every edge of the lattice can either take + or one of the 2 n colors labeledby [ ± n ].We denote the colors by c n , · · · , c n . Hereafter we refer to c i and c i as mutually opposite colors for every1 ≤ i ≤ n . For convenience of notations, we also let c := +. We take the following order on the colors(including c ): ¯ n < · · · < ¯1 < < < · · · n. (3.1)In the colored model, there are also two types of vertices. They are termed “colored stochastic Γ vertex”and “colored stochastic ∆ vertex”. The model depends on n spectral parameters z , · · · , z n and a deformationparameter q . Again we take z ′ i = q + 1 − z i . The Boltzmann weights for the two types of vertices are listed in Figures 18-19. a / a a / a b b c c c α c α c α c α z i c β c β c β c β z i c α c β c α c β z i c β c α c β c α z i c β c α c α c β z i c α c β c β c α z i z i qz i − qz i − z i Figure 18: Boltzmann weights for colored stochastic Γ vertex with spectral parameter z i , where α < β a / a a / a b b d d c α c α c α c α z i c β c β c β c β z i c α c β c α c β z i c β c α c β c α z i c β c β c α c α z i c α c α c β c β z i z ′ i q z ′ i − z ′ i − q z ′ i Figure 19: Boltzmann weights for colored stochastic ∆ vertex with spectral parameter z i , where α < β and z ′ i = q + 1 − z i The colored model consists of a rectangular lattice with 2 n rows and L columns. The rows are numbered1 , , · · · , n from bottom to top, and the columns are numbered 1 , , · · · , L from right to left. Every odd-numbered row is a row of colored stochastic ∆ vertex, and every even-numbered row is a row of coloredstochastic Γ vertex. The spectral parameter for the i th row of colored stochastic Γ vertices and the i th rowof the colored stochastic ∆ vertices is z i . 16he model also depends on a partition λ = ( λ , · · · , λ n ) ∈ Z n (with λ ≥ · · · ≥ λ n ) and two elements σ, τ ∈ B n , where B n is the hyperoctahedral group as introduced previously. We assume that L ≥ λ + n .The assignment of boundary conditions is given as follows: on the left column, we assign color c σ ( i ) to the i th row of colored stochastic Γ vertex, and + to each row of colored stochastic ∆ vertex; on the top, weassign + to each boundary edge; on the bottom, we assign color c τ ( i ) to each column labeled λ i + n + 1 − i for 1 ≤ i ≤ n , and assign + to the other columns. On the right, the i th row of colored stochastic Γ vertexand the i th row of colored stochastic ∆ vertex are connected with a cap. The Boltzmann weights for thecaps are given in Figure 20. Cap ++ c ¯ α c α c α c ¯ α Boltzmann weight 1 1 1Figure 20: Boltzmann weights of the caps for colored stochastic symplectic ice: where α ∈ { , , · · · , n } Hereafter we denote by S n,L,λ,σ,τ,z the collection of admissible configurations with the above specifieddata. We also denote by Z ( S n,L,λ,σ,τ,z ) the corresponding partition function.We note that the Boltzmann weights for both types of vertices and the caps are also stochastic, as inthe uncolored case. When z satisfies the condition (2.1), a probabilistic interpretation for each vertex cansimilarly be obtained.We also note that the colored model can be interpreted as an interacting particle system as in theuncolored case if condition (2.1) is satisfied. The interpretation is similar to the uncolored case, except thatnow each particle carries a color, and that the updating rule for the particle depends on its color.The detailed rule is as follows. When t is even, the particles are ordered from left to right. There is a newparticle entering from the left boundary with color c σ ( n − t ) (we call it 0th particle), which jumps to the rightwith geometric jump size (with parameter qz n − t if σ ( n − t ) >
0, or z n − t otherwise) unless it hits the 1stparticle; if the particle hits the 1st particle, the updating rule will be described later. Starting from l = 1, ifthe l th particle wasn’t hit by any particle on its left, we flip a coin with head probability z n − t (if the color ofthe particle is c α with α >
0) or qz n − t (if α <
0) to determine whether it will stay at its current position; ifthe coin comes up tail, then the particle jumps to the right with geometric jump size (with parameter qz n − t if α >
0, or z n − t if α <
0) unless it hits the ( l + 1)th particle; if the particle hits the ( l + 1)th particle, theupdating rule will be described later. If the l th particle was hit by the ( l − l − l th particle (depending on the updating rule as will be described later) jumps to the right by 1 andthe following move is the same as the previous case except for the first step determining whether it will stayat the current position. Then the ( l + 1)th particle begins to move. If the rightmost particle moves beyondthe first column (meaning that it hits the cap), it is reflected by the cap (meaning that it will start to moveleftward from the first column at time t + 1), and its color c α is changed to c α .When t is odd, the particles are ordered from right to left. If there is a particle reflected from the cap(we call it 0th particle), it jumps to the left with geometric jump size (with parameter q z ′ n − t − if the colorof the particle is c α with α >
0, or z ′ n − t − otherwise) unless it hits the 1st particle; if the particle hits the1st particle, the updating rule will be described later. Starting from l = 1, if the l th particle wasn’t hit byany particle on its right, we flip a coin with head probability z ′ n − t − (if the color of the particle is c α with α >
0) or q z ′ n − t − (if α <
0) to determine whether it will stay at its current position; if the coin comesup tail, then the particle jumps to the left with geometric jump size (with parameter q z ′ n − t − if α >
0, or z ′ n − t − if α <
0) unless it hits the ( l + 1)th particle; if the particle hits the ( l + 1)th particle, the updatingrule will be described later. If the l th particle was hit by the ( l − l − l thparticle (depending on the updating rule as will be described later) jumps to the left by 1 and the following17ove is the same as the previous case except for the first step determining whether it will stay at the currentposition. Then the ( l + 1)th particle begins to move.Now we describe the updating rule for the case when a particle hits another. When the time t is even,consider the situation when a particle of color c α hits another particle of color c β from the left. If α < β ,with probability z n − t the two particles are swapped, with the particle of color c β staying at the originalposition and the other particle continuing to move; with probability 1 − z n − t , the particle of color c α staysat the current position and the other particle starts to move. If α > β , with probability qz n − t the twoparticles are swapped, with the particle of color c β staying at the original position and the other particlecontinuing to move; with probability 1 − qz n − t , the particle of color c α stays at the current position andthe other particle starts to move.When the time t is odd, we also consider the situation when a particle of color c α hits another particle ofcolor c β from the right. If α < β , with probability z ′ n − t − the two particles are swapped, with the particleof color c β staying at the original position and the other particle continuing to move; with probability1 − z ′ n − t − , the particle of color c α stays at the current position and the other particle starts to move. If α > β , with probability q z ′ n − t − the two particles are swapped, with the particle of color c β staying at theoriginal position and the other particle continuing to move; with probability 1 − q z ′ n − t − , the particle ofcolor c α stays at the current position and the other particle starts to move.Under this probabilistic interpretation, the partition function Z ( S n,L,λ,σ,τ,z ) represents the probabilitythat (with the entering order of particle colors specified by σ ) the particle configuration at time t = 2 n isgiven by µ and τ , with µ specifying the particle locations and τ specifying the particle colors. For the colored stochastic symplectic ice, we find three sets of Yang-Baxter equations. The correspondingR-matrices are termed “colored stochastic Γ − Γ vertex”, “colored stochastic ∆ − Γ vertex” and “coloredstochastic ∆ − ∆ vertex”. In Section 3.5 we will show that, when combined with the reflection equation,these three sets of Yang-Baxter equations are enough for us to derive the recursive relations for the partitionfunctions.Throughout the paper we denote the Boltzmann weights of an R-matrix of type XY and spectral pa-rameters z i , z j as shown in Figure 21 by R XY ( c α , c β , c γ , c δ ; z i , z j ), where ( X, Y ) ∈ { (Γ , Γ) , (∆ , ∆) , (∆ , Γ) } and α, β, γ, δ ∈ { n, · · · , , , , · · · , n } . The Boltzmann weights for the three types of R-matrices are givenin Figures 22-24. c α c β c γ c δ R z i ,z j Figure 21: R-matrixThe following theorem gives the three sets of Yang-Baxter equations for the colored stochastic symplecticice.
Theorem 3.1.
For any ( X, Y ) ∈ { (∆ , Γ) , (Γ , Γ) , (∆ , ∆) } the following holds. Assume that S is coloredstochastic X vertex with spectral parameter z i , T is colored stochastic Y vertex with spectral parameter z j ,and R is colored stochastic X − Y vertex with spectral parameters z i , z j . Then the partition functions of the α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c α c α c β R z i ,z j c α c β c β c α R z i ,z j z i − z j − ( q +1) z j + qz i z j q ( z i − z j )1 − ( q +1) z j + qz i z j (1 − qz i )(1 − z j )1 − ( q +1) z j + qz i z j (1 − z i )(1 − qz j )1 − ( q +1) z j + qz i z j Figure 22: Boltzmann weights for colored stochastic Γ − Γ vertex with spectral parameters z i and z j : where α < βc α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c β c α c α R z i ,z j c α c α c β c β R z i ,z j z ′ i + qz j − ( q +1) z ′ i z j − z ′ i z j q − z ′ i + z j − (1+ q − ) z ′ i z j − z ′ i z j (1 − z ′ i )(1 − qz j )1 − z ′ i z j (1 − q − z ′ i )(1 − z j )1 − z ′ i z j Figure 23: Boltzmann weights for colored stochastic ∆ − Γ vertex with spectral parameters z i and z j : where α < β following two configurations are equal for any fixed combination of colors a, b, c, d, e, f ∈ { c ¯ n , · · · , c , · · · , c n } . ab c defg hi STR ab c def jk lTS R (3.2) Proof.
From conservation of colors for both colored stochastic Γ vertices and colored stochastic ∆ vertices(note that the directions of input and output are different for these two types of vertices), it can be checkedthat at most four distinct colors (including c ) can appear on the boundary edges in any of the two configu-rations, and that the color on an inner edge must be one of the colors on boundary edges (only consideringadmissible configurations). Moreover, the Boltzmann weight of a vertex only depends on the relative order ofthe colors on its adjacent four edges. Therefore it suffices to check the result for four colors, and there are atmost 4 possible combinations of boundary colors. These identities are checked using a SAGE program. σ ( i ) = τ ( i ) When σ ( i ) = τ ( i ) for every 1 ≤ i ≤ n , the partition function Z ( S n,L,λ,σ,τ,z ) has a relatively simple form, asis shown in the following theorem. 19 α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c α c α c β R z i ,z j c α c β c β c α R z i ,z j z ′ j − z ′ i q − ( q +1) z ′ i + z ′ i z ′ j q ( z ′ j − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ i )( q − z ′ j ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ j )( q − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j Figure 24: Boltzmann weights for colored stochastic ∆ − ∆ vertex with spectral parameters z i and z j : where α < β Theorem 3.2. If σ and τ satisfy the condition that σ ( i ) = τ ( i ) for every ≤ i ≤ n , then we have Z ( S n,L,λ,σ,τ,z ) = n Y i =1 z Li n Y i =1 ( z ′ i q ) λ i + n − i n Y i =1 (1 − q − σ ( i ) < z ′ i ) × q P ni =1 ( L − n + i + λ i )1 σ ( i ) > + P ≤ i We use the colored path interpretation. An illustration of the proof is shown in Figure 25, where weassume that c = B, c = R, c = B, c = R . We say that c i and c i are of the same color type for 1 ≤ i ≤ n (and c itself forms a color type). The collection of the particles with the same color type are viewed as acolored path. Each path enters from the left boundary, moves rightward or downward on each row of coloredstochastic Γ vertex, and moves leftward or downward on each row of colored stochastic ∆ vertex. When thepath enters the cap on a row of colored stochastic Γ vertex, it will bend through the cap, change the colorto its opposite, and restart on the right-most vertex of the row of colored stochastic ∆ vertex just below theprevious row. Finally, the colored path leaves the rectangular lattice at the bottom boundary.Consider the colored path entering from the 2nd row, which has color c σ (1) . In order for the path toleave the domain with an opposite color (which is required by the boundary condition, as τ (1) = σ (1)), ithas to move rightward until it goes through the cap. Then the path changes its color to c τ (1) and leaves thedomain at the column labeled as λ + n .Now consider the colored path entering from the 4th row, which has color c σ (2) . In order for the path toleave the domain with an opposite color, it has to move rightward until it goes through the cap (as the capconnecting the first two rows has already been taken by the colored path entering from the 2nd row). Thennote that λ + n > λ + n − 1. In order for the path to leave at the column labeled λ + n − 1, it has to moveleftward after passing the cap until it reaches the column labeled λ + n − 1. After that it moves downwarduntil it leaves the domain.The rest of the colored paths can be analyzed similarly. The colored path entering from the (2 i )th rowfirst moves rightward until it goes through the cap (and changes the color to its opposite), then it movesleftward until it reaches the column labeled λ i + n + 1 − i , and finally it moves downward until it leaves thedomain.The above analysis shows that there is only one admissible state. Computing the Boltzmann weight ofthis state finishes the proof. Due to the lack of the R-matrix and the Yang-Baxter equation for the colored stochastic Γ vertex andthe colored stochastic ∆ vertex, we cannot use the caduceus relation as in the uncolored model. However,20 z ∆ z Γ z ∆ z R R R R RRRR + + + ++ + + ++ + + R + + + R + B + R + + + ++ B B B B B + + B B B Theorem 3.3. Assume that S is colored stochastic Γ − Γ vertex of spectral parameters z i , z j , T is col-ored stochastic ∆ − Γ vertex of spectral parameters z i , z j , S ′ is colored stochastic ∆ − ∆ vertex of spec-tral parameters z j , z i , and T ′ is colored stochastic ∆ − Γ vertex of spectral parameters z j , z i . Denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) the partition function of the following configuration with fixed combination of colors ǫ , ǫ , ǫ , ǫ ∈ { c ¯ n , · · · , c , · · · , c n } . I = ǫ ǫ ǫ ǫ S T (3.5) Also denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) the partition function of the following configuration with fixed combination f colors ǫ , ǫ , ǫ , ǫ ∈ { c ¯ n , · · · , c , · · · , c n } . I = ǫ ǫ ǫ ǫ S ′ T ′ (3.6) Then for any fixed combination of colors ǫ , ǫ , ǫ , ǫ ∈ { c ¯ n , · · · , c , · · · , c n } , we have Z ( I ( ǫ , ǫ , ǫ , ǫ )) = Z ( I ( ǫ , ǫ , ǫ , ǫ )) . (3.7) Proof. We say that c α and c β are of the same color type, if β = ¯ α ( c itself forms a color type). Fromconservation of colors for the R-matrix and the cap weights, we can deduce that for any admissible state of I or I , each color type must appear for an even number of times in { ǫ , ǫ , ǫ , ǫ } , and that the color type ofan inner edge must be one of the color types of { ǫ , ǫ , ǫ , ǫ } . From this we can further deduce that at mosttwo color types can appear in { ǫ , ǫ , ǫ , ǫ } in any admissible state of I or I . Moreover, we note that theBoltzmann weight for the R-matrix only depends on the relative order of colors on its four adjacent edges.Therefore it suffices to check the relation for five colors { c ¯2 , c ¯1 , c , c , c } . There are at most 5 combinationsof ( ǫ , ǫ , ǫ , ǫ ) for this case. These identities have been checked using a SAGE program. In this section, we derive recursive relations of the partition function. The recursive relations are furtherrelated to Demazure-Lusztig operators of type C in Section 3.6. The main results are Theorems 3.4-3.5. Theorem 3.4. Assume that ≤ i ≤ n − and σ ( i + 1) > σ ( i ) . Let s i be the transposition ( i, i + 1) in B n ,and let s i z be the vector obtained from z by interchanging z i , z i +1 . Then the partition function of the coloredstochastic symplectic ice satisfies the following recursive relation: q σ ( i +1) > − σ ( i ) > Z ( S n,L,λ,σs i ,τ,z ) = − A ( q, z, i ) Z ( S n,L,λ,σ,τ,z ) + B ( q, z, i ) Z ( S n,L,λ,σ,τ,s i z ) , (3.8) where A ( q, z, i ) = (1 − z i +1 )(1 − qz i ) z i +1 − z i , (3.9) and B ( q, z, i ) = 1 − ( q + 1) z i + qz i z i +1 z i +1 − z i . (3.10) Theorem 3.5. Assume that σ ( n ) > . Let s n be the element of B n that changes the sign of the element atthe n th position, and s n z = ( z , · · · , z n − , z ′ n ) . (3.11) Then we have ( qz n ) L Z ( S n,L,λ,σs n ,τ,z ) = C ( q, z ) z − Ln Z ( S n,L,λ,σ,τ,z ) − D ( q, z ) z ′ Ln Z ( S n,L,λ,σ,τ,s n z ) . (3.12) where C ( q, z ) = ( q − z ′ n )( z n − q (1 − z n z ′ n ) , (3.13) D ( q, z ) = qz n + z ′ n − ( q + 1) z n z ′ n q (1 − z n z ′ n ) . (3.14)The rest of this section is devoted to the proof of Theorems 3.4-3.5.22 .5.1 Proof of Theorem 3.4 The proof of Theorem 3.4 is based on the Yang-Baxter equation (Theorem 3.1) and the reflection equation(Theorem 3.3). Proof of Theorem 3.4. We attach two R-vertices to the left of S n,L,λ,σ,τ,s i z , as shown in the following T ′ S ′ + c σ ( i +1) + c σ ( i ) z i +1 z i +1 z i z i ∆Γ∆Γ (3.15)where we omit the other rows of S n,L,λ,σ,τ,s i z , S ′ is colored stochastic ∆ − ∆ vertex of spectral parameters z i , z i +1 , and T ′ is colored stochastic ∆ − Γ R-vertex of spectral parameters z i , z i +1 . We denote by Z thepartition function of this new ice model.Note that the only admissible configuration of the two R-vertices is given as follows T ′ S ′ + c σ ( i +1) + c σ ( i ) ++ c σ ( i ) + z i +1 z i +1 z i z i ∆Γ∆Γ (3.16)Therefore we have Z = R ∆∆ (+ , + , + , +; z i , z i +1 ) R ∆ , Γ (+ , c σ ( i ) , + , c σ ( i ) ; z i , z i +1 ) Z ( S n,L,λ,σ,τ,s i z ) . (3.17)By Theorem 3.1, we can push the two R-vertices to the right without changing the partition function.Namely, we denote by Z the partition function of the following configuration+ c σ ( i ) + c σ ( i +1) z i z i +1 z i +1 z i ∆∆ΓΓ (3.18)Then we have Z = Z . 23y Theorem 3.3, Z is equal to the partition function of the following configuration+ c σ ( i ) + c σ ( i +1) z i z i +1 z i +1 z i ∆∆ΓΓ (3.19)Using Theorem 3.1, we push the two R-vertices back to the left without changing the partition function.Namely, if we denote the partition function of the following configuration by Z , then Z = Z . Here S iscolored stochastic Γ − Γ vertex of spectral parameters z i +1 , z i , and T is colored stochastic ∆ − Γ vertex ofspectral parameters z i +1 , z i . TS + c σ ( i ) + c σ ( i +1) z i z i z i +1 z i +1 ∆Γ∆Γ (3.20)Now we denote by Z and Z the partition functions of the following two configurations.++ c σ ( i ) c σ ( i +1) z i z i z i +1 z i +1 ∆Γ∆Γ ++ c σ ( i +1) c σ ( i ) z i z i z i +1 z i +1 ∆Γ∆Γ (3.21)By considering all possible configurations of S, T , we conclude that Z = R ΓΓ ( c σ ( i ) , c σ ( i +1) , c σ ( i +1) , c σ ( i ) ; z i +1 , z i ) R ∆Γ (+ , c σ ( i ) , + , c σ ( i ) ; z i +1 , z i ) Z + R ΓΓ ( c σ ( i ) , c σ ( i +1) , c σ ( i ) , c σ ( i +1) ; z i +1 , z i ) R ∆Γ (+ , c σ ( i +1) , + , c σ ( i +1) ; z i +1 , z i ) Z (3.22)Now note that Z = Z ( S n,L,λ,σ,τ,z ) , (3.23) Z = Z ( S n,L,λ,σs i ,τ,z ) . (3.24)Therefore we have R ∆∆ (+ , + , + , +; z i , z i +1 ) R ∆Γ (+ , c σ ( i ) , + , c σ ( i ) ; z i , z i +1 ) Z ( S n,L,λ,σ,τ,s i z )= R ΓΓ ( c σ ( i ) , c σ ( i +1) , c σ ( i +1) , c σ ( i ) ; z i +1 , z i ) R ∆Γ (+ , c σ ( i ) , + , c σ ( i ) ; z i +1 , z i ) Z ( S n,L,λ,σ,τ,z )+ R ΓΓ ( c σ ( i ) , c σ ( i +1) , c σ ( i ) , c σ ( i +1) ; z i +1 , z i ) R ∆Γ (+ , c σ ( i +1) , + , c σ ( i +1) ; z i +1 , z i ) Z ( S n,L,λ,σs i ,τ,z ) . (3.25)24sing the Boltzmann weights for colored R-matrices and simplifying the expressions, we obtain the conclusionof the theorem. The proof of Theorem 3.5 is based on the following idea. First note that only one color (other than c ),denoted by R , may appear in the 2 n th row. So we can simultaneously switch + and R in the 2 n th row andchange Boltzmann weights in a similar way as in the uncolored case. The recursive relation is then derivedfrom the Yang-Baxter equation (Theorem 3.1) and a variant of the fish relation using two auxiliary capsgiven by equation (3.32). Proof. We note that all the boundary edges on the top of the rectangular lattice carry the + spin, and thereare only two possible colors c σ ( n ) and + in the 2 n th row. We write R := c σ ( n ) hereafter to simplify thenotations. Therefore, only the three states in Figure 26 are involved in the 2 n th row of the lattice. Moreover,only the two states in Figure 27 are involved in the cap connecting the last two rows.+ + ++ z n R + R + z n R + + Rz n qz n − qz n Figure 26: Boltzmann weights involved in the 2 n th rowCap ++ RR Boltzmann weight 1 1Figure 27: Boltzmann weights involved for the cap connecting the last two rowsNow for each admissible state we change the color in the 2 n th row from + to R and from R to +(note that no other colors are involved in the 2 n th row for an admissible state). Meanwhile we change theBoltzmann weights for the vertices in the 2 n th row into the ones presented in Figure 28, and change theBoltzmann weights for the caps connecting the last two rows into those in Figure 29. Note that in theoriginal configuration, if the colored path entering from the left of the 2 n th row doesn’t go through thecap connecting the last two rows, then one c pattern is involved (and no c pattern is involved); otherwiseneither c nor c is involved. Thus noting the changed Boltzmann weights for the cap, we can deduce thatif we denote by Z the partition function of the new system, then Z = ( 1 qz n ) L Z ( S n,L,λ,σ,τ,z ) . (3.26)Note that the new weight for the 2 n th row is a colored stochastic ∆ vertex with spectral parameter z ′ n .Now we attach an R-vertex (colored stochastic ∆ − ∆ vertex with spectral parameters z ′ n and z n ) to the left25 / a a / a b b d d c α c α c α c α c β c β c β c β c α c β c α c β c β c α c β c α c β c β c α c α c α c α c β c β z n qz n − z n − qz n Figure 28: New Boltzmann weights for the 2 n th row, where α < β New cap R + + R Boltzmann weight 1 − z ′ n z n + (3.27)Note that the only admissible configuration of the R-matrix is given by++ ∆∆ z ′ n z n ++ ++ (3.28)Therefore, the partition function of the above system is equal to Z .By Theorem 3.1, we can push the R-vertex to the right without changing the partition function. Thatis, the partition function of the above system is equal to the partition function of the following++ ∆∆ z n z ′ n + (3.29)We introduce two types of auxiliary caps C and C . The weights for these caps are shown in Figures30-31.Now let Z ( I ( ǫ , ǫ )) be the partition function of the following system for every choice of ǫ , ǫ ∈{ c n , · · · , c n } (where the R-vertex is the one we used above, and the cap weights are given by those inFigure 29). I ( ǫ , ǫ ) = ǫ ǫ (3.30)26ew cap R + C + R C Boltzmann weight 1 − C New cap R + C + R C Boltzmann weight − C Also denote by Z ( I ( ǫ , ǫ )) , Z ( I ′ ( ǫ , ǫ )) the partition functions of the following two systems for every choiceof ǫ , ǫ ∈ { c n , · · · , c n } . I ( ǫ , ǫ ) = ǫ ǫ C and I ′ ( ǫ , ǫ ) = ǫ ǫ C (3.31)Then we can check that for any ǫ , ǫ , Z ( I ( ǫ , ǫ )) = ( qz n − − z ′ n ) qz n + z ′ n − ( q + 1) Z ( I ( ǫ , ǫ )) + q ( z n z ′ n − qz n + z ′ n − ( q + 1) Z ( I ′ ( ǫ , ǫ )) (3.32)Thus if we denote by Z , Z ′ the partition functions of the following two configurations, then++ ∆∆ z n z ′ n + C (3.33)++ ∆∆ z n z ′ n + C (3.34) Z = ( qz n − − z ′ n ) qz n + z ′ n − ( q + 1) Z + q ( z n z ′ n − qz n + z ′ n − ( q + 1) Z ′ . (3.35)Finally we compute Z , Z ′ . For Z , we change the Boltzmann weights of the 2 n th row to the weights inFigure 32. We also change the Boltzmann weight of the cap to those in Figure 33 (call it C ′ ). It can bechecked that the partition function doesn’t change.Now note that the top boundary edges all carry the + spin, we use the previous argument (changing R to + and + to R , and changing Boltzmann weights accordingly) to show that Z is equal to ( z ′ n q ) L times the27 / a a / a b b d d c α c α c α c α c β c β c β c β c α c β c α c β c β c α c β c α c β c β c α c α c α c α c β c β z ′ n q z ′ n z ′ n − q z ′ n − n th row, where α < β : for computing Z New cap R + C ′ + R C ′ Boltzmann weight 1 1Figure 33: Boltzmann weights for the cap C ′ partition function of the following system, which is Z ( S n,L,λ,σ,τ,s n z ).+ R Γ∆ z ′ n z ′ n + (3.36)For Z ′ , we change the Boltzmann weights of the 2 n th row to the weights in Figure 32. We also changethe Boltzmann weights of the cap to those in Figure 34 (call it C ′ ). It can be checked that the partitionfunction changes by a factor of − 1. Now note again that the top boundary edges all carry the + spin, weNew cap R + C ′ + R C ′ Boltzmann weight 1 1Figure 34: Boltzmann weights for the cap C ′ use the previous argument (this time we change R to + and + to R , and change the Boltzmann weightsaccordingly) to show that Z ′ is equal to − ( z ′ n ) L times the partition function of the following system, whichis Z ( S n,L,λ,σs n ,τ,s n z ). + R Γ∆ z ′ n z ′ n + (3.37)28herefore we conclude that( 1 z n ) L Z ( S n,L,λ,σ,τ,z ) = ( qz n − − z ′ n ) z ′ n + qz n − ( q + 1) z ′ Ln Z ( S n,L,λ,σ,τ,s n z ) − q ( z n z ′ n − qz n + z ′ n − ( q + 1) q L z ′ Ln Z ( S n,L,λ,σs n ,τ,s n z ) . (3.38)By rearranging and changing z to s n z we reach the conclusion of the theorem. The recursive relations for colored stochastic symplectic ice shown in Section 3.5 are related to Demazure-Lusztig operators of type C. We explain this connection in this section.Viewed as operators on rational functions of u = ( u , · · · , u n ), Demazure-Lusztig operators of type Ccan be given as follows (see [14],[27] for details). For 1 ≤ i ≤ n − 1, define s i ( u , · · · , u n ) = ( u , · · · , u i +1 , u i , · · · , u n ) , (3.39)that is, s i transposes u i and u i +1 . Also define s n ( u , · · · , u n ) = ( u , · · · , u n − , u − n ) . (3.40)For every 1 ≤ i ≤ n , and any rational function f ( u ) of u , we let s i f ( u ) := f ( s i u ) . (3.41)Then Demazure-Lusztig operators (with parameter v ) L i,v are given by L i,v ( f ) = 1 − vu α i − f + vu α i − u α i − s i ( f ) , (3.42)where { α i } are the simple roots of type C n , that is, α i = ǫ i − ǫ i +1 for 1 ≤ i ≤ n − α n = 2 ǫ n . Here ǫ i is the n -dimensional vector such that its i th coordinate is 1 and the other coordinates are 0, for every1 ≤ i ≤ n .We also let ˆ L i,v = L i,v − v + 1. Note that from the quadratic relation for L i,v L i,v = ( v − L i,v + v, (3.43)we obtain ˆ L i,v L i,v = v. (3.44)In order to relate the recursive relations to Demazure-Lusztig operator of type C, we make the followingchange of variables. We let u i = 1 − qz i − z i (3.45)for every 1 ≤ i ≤ n . Then we have z i = 1 − u i q − u i , (3.46) z ′ i = 1 − qu i − u i . (3.47)Under this change of variables, we obtain that for every 1 ≤ i ≤ n − A ( q, z, i ) = ( q − u i u i − u i +1 , (3.48)29 ( q, z, i ) = qu i − u i +1 u i − u i +1 , (3.49) C ( q, z ) = 1 − qq (1 − u n ) , (3.50) D ( q, z ) = 1 − qu n q (1 − u n ) , (3.51)where A ( q, z, i ) and B ( q, z, i ) are as in Theorem 3.4, and C ( q, z ) and D ( q, z ) are as in Theorem 3.5.Now we let Z ( ˜ S n,L,λ,σ,τ,u ) = Z ( S n,L,λ,σ,τ,z ) q P ni =1 ( n − i )1 σ ( i ) > +( L +1) P ni =1 σ ( i ) < Q ni =1 z Li (3.52)as a function of u = ( u , · · · , u n ). Then Theorems 3.4 and 3.5 translate into the following Theorem 3.6. For ≤ i ≤ n − , if σ ( i + 1) > σ ( i ) we have Z ( ˜ S n,L,λ,σs i ,τ,u ) = ˆ L i,q ( Z ( ˜ S n,L,λ,σ,τ,u )) . (3.53) Moreover, if σ ( n ) > , we have Z ( ˜ S n,L,λ,σs n ,τ,u ) = −L n,q ( Z ( ˜ S n,L,λ,σ,τ,u )) . (3.54) In this section, we present a different colored model for the stochastic symplectic ice. In this model, the setof colors is { c , · · · , c n } . We also denote by c = +.The model and related Boltzmann weights are introduced in Section 4.1. Then the Boltzmann weightsfor the R-matrices are introduced in Section 4.2, and the Yang-Baxter equation is proved there. Finallyin Section 4.3 the reflection equation is introduced, based on which the recursive relations of the partitionfunction are derived. We introduce the new colored stochastic symplectic ice in this section. The set of colors for this model isgiven by { c , · · · , c n } .In this model, there are also two types of vertices termed “colored stochastic Γ vertex” and “coloredstochastic ∆ vertex”. The model depends on n spectral parameters z , · · · , z n and a deformation parameter q , and we take z ′ i = q + 1 − z i . The Boltzmann weights for these two types of vertices are listed in Figures 35-36. a / a a / a b b c c c α c α c α c α z i c β c β c β c β z i c α c β c α c β z i c β c α c β c α z i c β c α c α c β z i c α c β c β c α z i z i qz i − qz i − z i Figure 35: Boltzmann weights for colored stochastic Γ vertex with spectral parameter z i , where α < β / a a / a b b d d c α c α c α c α z i c β c β c β c β z i c α c β c α c β z i c β c α c β c α z i c β c β c α c α z i c α c α c β c β z i z ′ i q z ′ i − z ′ i − q z ′ i Figure 36: Boltzmann weights for colored stochastic ∆ vertex with spectral parameter z i , where α < β and z ′ i = q + 1 − z i The basic set-up of the new model is similar to that of the colored model given in Section 3. The differencelies in the assignment of boundary conditions: now we specify two permutations σ, τ from the symmetricgroup S n , assigning the color c τ ( i ) to each column labeled λ i + n + 1 − i for 1 ≤ i ≤ n at the bottom, andassigning the color c σ ( i ) to the left boundary of the i th row of colored stochastic Γ vertex. The Boltzmannweights for the caps are given in Figure 37.Cap ++ c α c α Boltzmann weight 1 1Figure 37: Boltzmann weights of the caps for the new colored model, where α ∈ { , , · · · , n } Hereafter we denote by U n,L,λ,σ,τ,z the collection of admissible configurations with the correspondingdata. We also denote by Z ( U n,L,λ,σ,τ,z ) the partition function. We assume that L ≥ λ + n , too.We note that the Boltzmann weights for this model are also stochastic, which allows a probabilistic inter-pretation of the model when the condition (2.1) is satisfied. The colored model can be similarly interpretedas stochastic dynamics as the colored model given in Section 3. The main difference is that in this model,the particles don’t change color when they are reflected at the caps. For this model, we also find three sets of Yang-Baxter equations. The corresponding R-matrices are termed“colored stochastic Γ − Γ vertex”, “colored stochastic ∆ − Γ vertex” and “colored stochastic ∆ − ∆ vertex”,too. The Boltzmann weights for the three types of R-matrices are given in Figures 38-40. c α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c α c α c β R z i ,z j c α c β c β c α R z i ,z j z i − z j − ( q +1) z j + qz i z j q ( z i − z j )1 − ( q +1) z j + qz i z j (1 − qz i )(1 − z j )1 − ( q +1) z j + qz i z j (1 − z i )(1 − qz j )1 − ( q +1) z j + qz i z j Figure 38: Boltzmann weights for colored stochastic Γ − Γ vertex with spectral parameters z i and z j : where α < β α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c β c α c α R z i ,z j c α c α c β c β R z i ,z j z ′ i + qz j − ( q +1) z ′ i z j − z ′ i z j q − z ′ i + z j − (1+ q − ) z ′ i z j − z ′ i z j (1 − z ′ i )(1 − qz j )1 − z ′ i z j (1 − q − z ′ i )(1 − z j )1 − z ′ i z j Figure 39: Boltzmann weights for colored stochastic ∆ − Γ vertex with spectral parameters z i and z j : where α < β c α c α c α c α R z i ,z j c β c β c β c β R z i ,z j c α c β c α c β R z i ,z j c β c α c β c α R z i ,z j c β c α c α c β R z i ,z j c α c β c β c α R z i ,z j z ′ j − z ′ i q − ( q +1) z ′ i + z ′ i z ′ j q ( z ′ j − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ i )( q − z ′ j ) q − ( q +1) z ′ i + z ′ i z ′ j (1 − z ′ j )( q − z ′ i ) q − ( q +1) z ′ i + z ′ i z ′ j Figure 40: Boltzmann weights for colored stochastic ∆ − ∆ vertex with spectral parameters z i and z j : where α < β The following theorem gives the three sets of Yang-Baxter equations for the colored stochastic symplecticice. Theorem 4.1. For any ( X, Y ) ∈ { (∆ , Γ) , (Γ , Γ) , (∆ , ∆) } the following holds. Assume that S is coloredstochastic X vertex with spectral parameter z i , T is colored stochastic Y vertex with spectral parameter z j ,and R is colored stochastic X − Y vertex with spectral parameters z i , z j . Then the partition functions of thefollowing two configurations are equal for any fixed combination of colors a, b, c, d, e, f ∈ { c , · · · , c n } . ab c defg hi STR ab c def jk lTS R (4.1) Proof. From conservation of colors for both colored stochastic Γ vertices and colored stochastic ∆ vertices(note that the directions of input and output are different for these two types of vertices), it can be checkedthat at most four distinct colors (including c ) can appear on the boundary edges in any of the two configu-rations, and that the color on an inner edge must be one of the colors on boundary edges (only consideringadmissible configurations). Moreover, the Boltzmann weight of a vertex only depends on the relative order ofthe colors on its adjacent four edges. Therefore it suffices to check the result for four colors, and there are atmost 4 possible combinations of boundary colors. These identities are checked using a SAGE program.32 .3 The reflection equation and recursive relations of the partition function In the new model, we also have the reflection equation: Theorem 4.2. Assume that S is colored stochastic Γ − Γ vertex of spectral parameters z i , z j , T is coloredstochastic ∆ − Γ vertex of spectral parameters z i , z j , S ′ is colored stochastic ∆ − ∆ vertex of spectral parameters z j , z i , and T ′ is colored stochastic ∆ − Γ vertex of spectral parameters z j , z i . Denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) thepartition function of the following configuration with fixed combination of colors ǫ , ǫ , ǫ , ǫ ∈ { c , · · · , c n } . I = ǫ ǫ ǫ ǫ S T (4.2) Also denote by Z ( I ( ǫ , ǫ , ǫ , ǫ )) the partition function of the following configuration with fixed combinationof colors ǫ , ǫ , ǫ , ǫ ∈ { c , · · · , c n } . I = ǫ ǫ ǫ ǫ S ′ T ′ (4.3) Then for any fixed combination of colors ǫ , ǫ , ǫ , ǫ ∈ { c , · · · , c n } , we have Z ( I ( ǫ , ǫ , ǫ , ǫ )) = Z ( I ( ǫ , ǫ , ǫ , ǫ )) . (4.4) Proof. From conservation of colors for the R-matrix and the cap weights, we can deduce that for anyadmissible state of I or I , each color must appear for an even number of times in { ǫ , ǫ , ǫ , ǫ } , and thatthe color of an inner edge must be one of the colors of { ǫ , ǫ , ǫ , ǫ } . From this we can further deduce that atmost two colors can appear in { ǫ , ǫ , ǫ , ǫ } in any admissible state of I or I . Moreover, we note that theBoltzmann weight for the R-matrix only depends on the relative order of colors on its four adjacent edges.Therefore it suffices to check the relation for three colors { c , c , c } . There are at most 3 combinations of( ǫ , ǫ , ǫ , ǫ ) for this case. These identities have been checked using a SAGE program.Based on the Yang-Baxter equation (Theorem 4.1) and the reflection equation (Theorem 4.2), we canestablish the following theorem on the recursive relations for the partition function Z ( U n,L,λ,σ,τ,z ). The proofof Theorem 4.3 is similar to that of Theorem 3.4 and we omit it. Theorem 4.3. Assume that ≤ i ≤ n − and σ ( i + 1) > σ ( i ) . Let s i be the transposition ( i, i + 1) in thesymmetric group S n , and let s i z be the vector obtained from z by interchanging z i , z i +1 . Then the partitionfunction of the new colored model satisfies the following recursive relation: Z ( U n,L,λ,σs i ,τ,z ) = − A ( q, z, i ) Z ( U n,L,λ,σ,τ,z ) + B ( q, z, i ) Z ( U n,L,λ,σ,τ,s i z ) , (4.5) where A ( q, z, i ) = (1 − z i +1 )(1 − qz i ) z i +1 − z i , (4.6) and B ( q, z, i ) = 1 − ( q + 1) z i + qz i z i +1 z i +1 − z i . (4.7)33 eferences [1] Baxter, R. 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