Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for A^{(1)}_n

We study an integrable vertex model with a periodic boundary condition associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.


Background
In [25], a class of soliton cellular automata (SCA) on a periodic one dimensional lattice was introduced associated with the non-exceptional quantum affine algebras U q (g n ) at q = 0. In this paper we focus on the type g n = A (1) n . The case n = 1 was introduced earlier as the periodic box-ball system [41]. It is a periodic version of Takahashi-Satsuma's soliton cellular automaton [38] originally defined on the infinite lattice. In the pioneering work [40], a closed formula for the dynamical period of the periodic box-ball system was found for the states with a trivial internal symmetry.
After some while, the first complete solution of the initial value problem and its explicit formula by a tropical analogue of Riemann theta function were obtained in [24,20,21]. This was done by developing the inverse scattering/spectral method 1 [6] in a tropical (ultradiscrete) setting primarily based on the quantum integrability of the system. It provides a linearization scheme of the nonlinear dynamics into straight motions. The outcome is practical as well as conceptual. For instance practically, a closed formula of the exact dynamical period for arbitrary states [24,Theorem 4.9] follows straightforwardly without a cumbersome combinatorial argument, see (3.40). Perhaps what is more important conceptually is, it manifested a decent tropical analogue of the theory of quasi-periodic soliton solutions [3,4] and its quantum aspects in the realm of SCA. In fact, several essential tools in classical integrable systems have found their quantum counterparts at a combinatorial level. Here is a table of rough correspondence. classical quantum solitons Bethe strings discrete KP/Toda flow fusion transfer matrix action-angle variable extended rigged configuration mod (q = 0 Bethe eq.) Abel-Jacobi map, Jacobi inversion modified Kerov-Kirillov-Reshetikhin bijection Riemann theta function charge of extended rigged configuration Here the Kerov-Kirillov-Reshetikhin (KKR) bijection [15,16] is the celebrated algorithm on rigged configurations in the combinatorial Bethe ansatz explained in Appendix C. As for the last line, see (1.1).
In a more recent development [36], a complete decomposition of the phase space of the periodic box-ball system has been accomplished into connected components under the time evolutions, and each of them is identified with the set of integer points on a certain torus Z g /F Z g explicitly.

A quick exposition
The aim of this paper is to extend all these results [24,20,21,36] to a higher rank case based on several conjectures. The results will be demonstrated with a number of examples. We call the system the periodic A (1) n soliton cellular automaton (SCA). It is a dynamical system on the L numbers {1, 2, . . . , n + 1} L which we call paths. There is the n-tuple of commuting series of time evolutions T (1) l , . . . , T (n) l with l ≥ 1. The system is periodic in that the cyclic shift (2.3) is contained in the member as T (1) 1 . The casual exposition in the following example may be helpful to grasp the idea quickly.  Note also that the first Young diagram (33222) gives the list of amplitudes of solitons. The two Young diagrams are conserved quantities (action variable). A part of a Young diagram having equal width is called a block. There are 4 blocks containing 2, 3, 1, 1 rows in this example. The 7 dimensional vector consisting of the listed numbers is the angle variable flowing linearly. It should be understood as an element of (Hint: (4.17)) where S m is the symmetric group of degree m. We have exhibited the trivial S 1 's as another hint. The matrix B is given by (4.11). The description is simplified further if one realizes that the relative (i.e., differences of) components within each block remain unchanged throughout. Thus picking the bottom component only from each block, we obtain where F ′ = F · diag( 1 2 , 1, 1, 1). Here the two kinds of tori arise reflecting that the two kinds of internal symmetries are allowed for the pair ((33222), (41)) under consideration.

Related works
A few remarks are in order on related works.
1. The solution of initial value problem of the n = 1 periodic box-ball system [24,20,21] has been reproduced partially by the procedure called 10-elimination [29]. By now the precise relation between the two approaches has been shown [17]. It is not known whether the 10elimination admits a decent generalization. The KKR bijection on the other hand is a canonical algorithm allowing generalizations not only to type A (1) n [18] but also beyond [32]. Conjecturally the approach based on the KKR type bijection will work universally for the periodic SCA [25] associated with Kirillov-Reshetikhin crystals as exemplified for A (1) 1 higher spin case [22] and A (1) n [25, 26, this paper]. In fact, the essential results like torus, dynamical period and phase space volume formula are all presented in a universal form by the data from Bethe ansatz and the root system 2 . From a practical point of view, it should also be recognized that the KKR map φ ±1 is a delightfully elementary algorithm described in less than half a page in our setting in Appendix C. It is a useful exercise to program it to follow examples in this paper.
2. When there is no duplication of the amplitudes of solitons, the tropical analogue of the Jacobian J (µ) obtained in [24] has an interpretation from the tropical geometry point of view [9,30]. In this paper we are naturally led to a higher rank version of J (µ) and the relevant tropical analogue of the Riemann theta function. We will call them 'tropical . . .' rather casually without identifying an underlying tropical geometric objects hoping not to cause a too much embarrassment. 3. In [28], the dynamical system on B 1,l 1 ⊗ · · · ⊗ B 1,l L equipped with the unique time evolution T Our periodic A (1) n SCA is the generalization of the generalized periodic box-ball system with ∀ l i = 1 that is furnished with the commuting family of time evolutions {T (r) l | 1 ≤ r ≤ n, l ≥ 1}. As we will see in the rest of the paper, it is crucial to consider this wider variety of dynamics and their whole joint spectrum {E (r) l }. They turn out to be the necessary and sufficient conserved quantities to formulate the inverse scattering method. Such a usage of the full family {T (r) l } was firstly proposed in the periodic setting in [25,26]. In particular in the latter reference, the most general periodic A (1) n SCA on B r 1 ,l 1 ⊗ · · · ⊗ B r 1 ,l L endowed with the dynamics {T (r) l } was investigated, and the dynamical period and a phase space volume formula were conjectured using some heuristic connection with combinatorial Bethe ansatz. This paper concerns the basic case ∀ r k = ∀ l k = 1 only but goes deeper to explore the linearization scheme under which the earlier conjectures [25,26] get refined and become simple corollaries. We expect that essential features in the inverse scattering formalism are not too much influenced by the choice of {r k , l k }.

Contents of paper
Let us digest the contents of the paper along Sections 2-4.
In Section 2, we formulate the periodic A (1) n SCA. It is a solvable vertex model [1] associated with the quantum affine algebra U q (A (1) n ) at q = 0 in the sense that notions concerning U q (A (1) n ) are replaced by those from the crystal theory [14,12,13], a theory of quantum group at q = 0. The correspondence is shown between the first and the second columns of the following table, where we hope the third column may be more friendly to the readers not necessarily familiar with the crystal theory.
The definition of the Kirillov-Reshetikhin module was firstly given in [19, Definition 1.1], although we do not use it in this paper. As mentioned previously, there is the n-tuple of commuting series of time evolutions T (1) l , . . . , T (n) l with l ≥ 1. The first series T (1) l is the ultradiscrete Toda/KP flow [7,39], and especially its top T (1) ∞ admits a simple description by a ball-moving algorithm like the periodic box-ball system. See Theorem 2.1 and comments following it. However, as again said previously, what is more essential in our approach is to make use of the entire family T It is the totality of the joint spectrum E (r) l that makes it possible to characterize the level set P(µ) (2.11) by the n-tuple of Young diagrams and the whole development thereafter.
As an illustration, T  2  4  1  1  2  3  4  3  1  12  34  12  23  13  24  11  24  11  24  11  22  12  23  13  24  23  34  12  34  4  2  3  1  4  1  2  1  3 by using the 'hidden variable' called carrier on horizontal edges belonging to B 2,2 . This is a conventional diagram representing the row transfer matrix of a vertex model [1] whose auxiliary (horizontal) space has the 'fusion type' B 2,2 . The general case T (r) l is similarly defined by using B r,l . The peculiarity as a vertex model is that there is no thermal fluctuation due to the crystallizing choice q = 0 resulting in a deterministic dynamics. Note that the carrier has been chosen specially so that the leftmost and the rightmost ones coincide to match the periodic boundary condition. This non-local postulate makes the well-definedness of the dynamics highly nontrivial and is in fact a source of the most intriguing features of our SCA. It is an important problem to characterize the situation in which all the time evolutions act stably. With regard to this, we propose a neat sufficient condition in (2.14) under which we will mainly work. See Conjecture 2.1. Our most general claim is an elaborate one in Conjecture 3.4. In Section 2.3, we explain the rigged configurations and Kerov-Kirillov-Reshetikhin bijection which are essential tools from the combinatorial Bethe ansatz [15,16] at q = 1. Based on Theorem 2.2, we identify the solitons and strings in (2.19).
In Section 3, we present the inverse scattering formalism, the solution algorithm of the initial value problem together with applications to the volume formula of the phase space and the dynamical period. The content is based on a couple of conjectures whose status is summarized in Section 3.8. Our strategy is a synthesis of the combinatorial Bethe ansätze at q = 1 [15,16] and q = 0 [19], or in other words, a modification of the KKR theory to match the periodic boundary condition. The action-angle variables are constructed from the rigged configurations invented at q = 1 by a quasi-periodic extension (3.1) followed by an identification compatible with the q = 0 Bethe equation (3.42). More concretely, the action variable is the n-tuple of Young diagrams µ = (µ (1) , . . . , µ (n) ) preserved under time evolutions. The angle variables live in a tropical analogue of the Jacobi-variety J (µ) and undergo a straight motion with the velocity corresponding to a given time evolution. Roughly speaking, the action and angle variables represent the amplitudes and the phases of solitons, respectively as demonstrated in Example 1.1. The modified KKR bijection Φ, Φ −1 yield the direct and inverse scattering maps. Schematically these aspects are summarized in the commutative diagram (Conjecture 3.2): where T stands for the commuting family of time evolutions T (r) l . Since its action on J (µ) is linear, this diagram achieves the solution of the initial value problem conceptually. Practical calculations can be found in Examples 3.2 and 3.5.
In Section 3.3 we decompose the level set P(µ) further into connected components, i.e., Torbits, and identify each of them with Z g /F γ Z g , the set of integer points on a torus with g and F γ explicitly specified in (2.9) and (3.25). Here γ denotes the order of symmetry in the angle variables. As a result we obtain (Theorem 3.1) The first equality follows from the decomposition into tori and the second one is a slight calculation. The last expression was known as the number of Bethe roots at q = 0 [19]. Thus this identity offers the Bethe ansatz formula a most elegant interpretation by the structure of the phase space of the periodic SCA.
Once the linearization scheme is formulated, it is straightforward to determine the dynamical period, the smallest positive integer N satisfying T N (p) = p for any time evolution T ∈ T and path p ∈ P(µ). The result is given in Theorem 3.2 and Remark 3.4. We emphasize that (3.39) is a closed formula that gives the exact (not a multiple of) dynamical period even when there are more than one solitons with equal amplitudes and their order of symmetry γ is nontrivial. See Example 3.7 in this paper for n = 2 and also [24,Example 4.10] for n = 1. In Section 3.6, we explain the precise relation of our linearization scheme to the Bethe ansatz at q = 0 [19]. The angle variables are actually in one to one correspondence with what we call the Bethe root at q = 0. These results are natural generalizations of the n = 1 case proved in [24,36]. In Section 3.7, we treat the general case (3.46) relaxing the condition (2.14). It turns out that the linearization scheme remains the same provided one discards some time evolutions and restricts the dynamics to a subgroup T ′ of T . This uncovers a new feature at n > 1.
In Section 4, we derive an explicit formula for the path p ∈ P(µ) that corresponds to a given action-angle variable (Theorem 4.3). This is a tropical analogue of the Jacobi inversion problem, and the result is indeed expressed by a tropical analogue of the Riemann theta function (4.14): Here the G × G period matrix B is specified by (4.6) from the n-tuple of Young diagrams µ as in (2.5)-(2.9). Theorems 4.2 and 4.3 are derived from the explicit formula of the KKR map by the tropical tau function τ trop for A (1) n [23]. It is a piecewise linear function on a rigged configuration related to its charge. The key to the derivation is the identity first discovered in [20] for n = 1. Here the limit sends the rigged configuration RC into the infinitely large rigged configuration obtained by the quasi-periodic extension. See (4.13) for the precise form. It is known [23] that τ trop is indeed the tropical analogue (ultradiscrete limit) of a tau function in the KP hierarchy [11]. Thus our result can be viewed as a fermionization of the Bethe ansatz and quasi-periodic solitons at a combinatorial level. As applications, joint eigenvectors of T (r) l are constructed that possess every aspect as the Bethe eigenvectors at q = 0 (Section 4.2), the dynamical period is linked with the q = 0 Bethe eigenvalue (Section 4.3), and miscellaneous calculations of some time average are presented (Section 4.4).
The main text is followed by 4 appendices. Appendix A recalls the row and column insertions following [5] which is necessary to understand the rule (2.1) that governs the local dynamics. Appendix B contains a proof of Theorem 2.1 based on crystal theory. Appendix C is a quickest and self-contained exposition of the algorithm for the KKR bijection. Appendix D is a proof of Theorem 2.2 using a result by Sakamoto [33] on energy of paths. n . There is a canonical bijection, the isomorphism of crystals, between B r,l ⊗B 1,1 and B 1,1 ⊗B r,l called combinatorial R. It is denoted either by

Periodic
R is uniquely determined by the condition that the product tableaux c · b and b ′ · c ′ coincide [34]. Here, the product c · b for example signifies the column insertion of c into b, which is also obtained by the row insertion of b into c [5]. Since c and c ′ are single numbers in our case, it is simplest to demand the equality See Appendix A for the definitions of the row and column insertions. The insertion procedure also determines an integer H(b ⊗ c) = H(c ′ ⊗ b ′ ) called local energy. We specify it as H = 0 or 1 according as the shape of the common product tableau is ((l + 1), l r−1 ) or (l r , 1), respectively 3 . Note that R and H refer to the pair (r, l) although we suppress the dependence on it in the notation. We depict the relation We will often suppress e. The horizontal and vertical lines here carry an element from B r,l and B 1,1 , respectively. We remark that R is trivial, namely R(b ⊗ c) = b ⊗ c, for (r, l) = (1, 1) by the definition.

Definition of dynamics
Fix a positive integer L and set B = (B 1,1 ) ⊗L . An element of B is called a path. A path b 1 ⊗ · · · ⊗ b L will often be written simply as a word b 1 b 2 . . . b L . Our periodic A (1) n soliton cellular automaton (SCA) is a dynamical system on a subset of B. To define the time evolution T (r) l associated with B r,l , we consider a bijective map B r,l ⊗ B → B ⊗ B r,l and the local energy obtained by repeated use of R. Schematically, the map and the local energy e 1 , . . . , e L are defined by the composition of the previous diagram as follows.
as the functions of v containing p as a 'parameter'. Naively, we wish to define the time evolution T (r) l of the path p as T (r) l (p) = p ′ by using a carrier v that satisfies the periodic boundary condition v = v ′ (v; p). This idea indeed works without a difficulty for n = 1 [24] 4 . In addition, T (1) 1 is always well defined for general n, yielding the cyclic shift: This is due to the triviality of R on B 1,1 ⊗ B 1,1 mentioned above. In fact the unique carrier is specified as v = v ′ = b L . Apart from this however, one encounters the three problems (i)-(iii) to overcome in general for n ≥ 2. For some p ∈ B, one may suffer from (i) Non-existence. There may be no carrier v satisfying v = v ′ (v; p).
(ii) Non-uniqueness. There may be more than one carriers, To cope with these problems, we introduce the notion of evolvability of a path according to [26].
, and moreover p ′ = p ′ (v; p), e k = e k (v; p) are unique for possibly non-unique choice of v. 5 In this case we define indicating that the time evolution operator T l -evolvable for all r, l, then it is simply called evolvable. The third problem in defining the dynamics is In fact, the non-uniqueness problem (ii) takes place as follows: . Let µ = (µ (1) , . . . , µ (n) ) be an n-tuple of Young diagrams. From µ (a) , we specify the data m where (C ab ) 1≤a,b≤n is the Cartan matrix of A n , i.e., C ab = 2δ ab − δ a,b+1 − δ a,b−1 . The quantity p (a) i is called a vacancy number. 6 The meaning of i in m here has been altered(!) from the literatures [19,20,21,22,23,24,25,26]. When the space is tight, we will often write (aiα) for (a, i, α) ∈ H and (ai) for (a, i) ∈ H, etc.
Given µ = (µ (1) , . . . , µ (n) ), we define the sets of paths B ⊃ P ⊃ P(µ) ⊃ P + (µ) by P = {p ∈ B | #(1) ≥ #(2) ≥ · · · ≥ #(n + 1)}, (2.10) where #(a) is the number of a ∈ B 1,1 in p = b 1 ⊗ · · · ⊗ b L . In view of the Weyl group symmetry [26,Theorem 2.2], and the obvious property that T (r) l is weight preserving, we restrict ourselves to P which is the set of paths with nonnegative weight. A path b 1 ⊗ · · · ⊗ b L is highest if the prefix b 1 ⊗ · · · ⊗ b j satisfies the condition #(1) ≥ #(2) ≥ · · · ≥ #(n + 1) for all 1 ≤ j ≤ L. We call P(µ) the level set associated with µ. The n-tuple µ of Young diagrams or equivalently the data (m i ) (ai)∈H will be referred to as the soliton content of P(µ) or the paths contained in it. This terminology comes from the fact that when L is large and #(1) ≫ #(2), . . . , #(n+1), it is known that the paths in P(µ) consist of m  Here a soliton with amplitude l is a part of a path of the form j 1 ⊗ j 2 ⊗ · · · ⊗ j l ∈ (B 1,1 ) ⊗l with j 1 ≥ · · · ≥ j l ≥ 2 by regarding 1 ∈ B 1,1 as an empty background. Note that a soliton is endowed not only with the amplitude l but also the internal degrees of freedom {j i }. The data m Note that the relation in (2.11) can be inverted as (E (a) otherwise. (2.13) Therefore the correspondence between the soliton content µ = µ (1) , . . . , µ (n) and the energy {E . Now we are ready to propose a sufficient condition under which the annoying feature in problem (iii) is absent: (2.14) This condition was first introduced in [22].  l (P(µ)) = P(µ) implies that one can apply any time evolution in any order for arbitrary times on any path p ∈ P(µ). In particular the inverse T (r) l −1 exists. We denote by Σ(p) the set of all paths generated from p in such a manner. By the definition it is a subset of the level set P(µ). Call Σ(p) the connected component of the level set containing the path p.
One can give another characterization of Σ(p) in terms of group actions. Let T be the abelian group generated by all T (r) l 's. Then T acts on the level set P(µ). This group action is not transitive in general, i.e., P(µ) is generically decomposed into several T -orbits. In what follows, connected components and T -orbits mean the same thing.  1)). The vacancy numbers are given as p ≥1 (X i ) = X i+3 and the same relations for Y i . Hence the claim of Conjecture 2.1 is valid for this P(µ).

The time evolution T
(1) ∞ has an especially simple description, which we shall now explain. Let B 1 be the set of paths in which 1 ∈ B 1,1 is contained most. Namely, Thus we see P ⊂ B 1 ⊂ B. We define the weight preserving map K a : B 1 → B 1 for a = 2, 3, . . . , n + 1. Given a path p ∈ B 1 regarded as a word p = b 1 b 2 . . . b L ∈ {1, . . . , n + 1} L , the image K a (p) is determined by the following procedure.
(i) Ignore all the numbers except 1 and a.
(ii) Connect every adjacent pair a1 (not 1a) by an arc, where 1 is on the right of a cyclically.
(iii) Repeat (ii) ignoring the already connected pairs until all a's are connected to some 1.
(iv) Exchange a and 1 within each connected pair.
∞ is factorized as follows: A proof is available in Appendix B, where each K a is obtained as a gauge transformed simple reflection for type A (1) n affine Weyl group. The dynamics of the form (2.16) has an interpretation as the ultradiscrete Toda or KP flow [7,39]. See also [28,Theorem VI.2]. As an additional remark, a factorized time evolution similar to (2.16) or equivalently (B.4) has been formulated for a periodic SCA associated to any non-exceptional affine Lie algebras [25]. In the infinite (non-periodic) lattice case, it was first invented by Takahashi [37] for type A (1) n . The origin of such a factorization is the factorization of the combinatorial R itself in a certain asymptotic domain, which has been proved uniformly for all non-exceptional types in [8]. The procedure (i)-(iv) is delightfully simple but inevitably non-local as the result of expelling the carrier out from the description. It is an interesting question whether the other typical time evolutions T ∞ admit a similar description without a carrier.

Rigged configuration
An n-tuple of Young diagrams µ = (µ (1) , . . . , µ (n) ) satisfying p (a) i ≥ 0 for all (a, i) ∈ H is called a configuration. Thus, those µ satisfying (2.14) form a subset of configurations. Consider the configuration µ attached with the integer arrays called rigging r = (r (a) ) 1≤a≤n = (r is satisfied 7 . We let RC(µ) denote the set of rigged configurations whose configuration is µ. It is well known [15,16] that the highest paths in B are in one to one correspondence with the rigged configurations by the Kerov-Kirillov-Reshetikhin (KKR) map where the union is taken over all the configurations. The both φ and φ −1 can be described by an explicit algorithm as described in Appendix C. Our main claim in this subsection is the following, which is a refinement of (2.18) with respect to µ and an adaptation to the periodic setting.
Theorem 2.2. The restriction of the KKR map φ to P + (µ) separates the image according to µ: The proof is available in Appendix D. We expect that this restricted injection is still a bijection but we do not need this fact in this paper.
Let us explain some background and significance of Theorem 2.2. Rigged configurations were invented [15] as combinatorial substitutes of solutions to the Bethe equation under string hypothesis [2]. The KKR map φ −1 is the combinatorial analogue of producing the Bethe vector from the solutions to the Bethe equation. The relevant integrable system is sl n+1 Heisenberg chain, or more generally the rational vertex model associated with U q A (1) n at q = 1. In this context, the configuration µ (2.5) is the string content specifying that there are m (a) i strings 7 We do not include L in the definition of rigged configuration understanding that it is fixed. It is actually necessary to determine the vacancy number p with color a and length l (a) i . Theorem 2.2 identifies the two meanings of µ. Namely, the soliton content for P(µ) measured by energy is equal to the string content for RC(µ) determined by the KKR bijection. Symbolically, we have the identity soliton = string, (2.19) which lies at the heart of the whole combinatorial Bethe ansatz approaches to the soliton cellular automata [20,21,22,23,24,25,26]. It connects the energy and the configuration by (2.13), and in a broader sense, crystal theory and Bethe ansatz. Further arguments will be given around (4.23). For n = 1, Theorem 2.2 has been obtained in [24,Proposition 3.4]. In the sequel, we shall call the n-tuple of Young diagrams µ = (µ (1) , . . . , µ (n) ) either as soliton content, string content or configuration when p (a) where we set |µ (0) | = L and |µ (n+1) | = 0. This is due to a known property of the KKR map φ and the fact that µ is a conserved quantity and Conjecture 3.5.
1,1 ≤ 1 and 0 ≤ r Hence there are 2 · 2 · 10 = 40 rigged configurations in RC(µ). The elements of P + (µ) and the riggings for their images under the KKR map φ are given in the following Here the riggings denote r 1,1 .

Action-angle variables
By action-angle variables for the periodic A n SCA, we mean the variables or combinatorial objects that are conserved (action) or growing linearly (angle) under the commuting family of time evolutions {T (r) l }. They are scattering data in the context of inverse scattering method [6,3,4]. In our approach, the action-angle variables are constructed by a suitable extension of rigged configurations, which exploits a connection to the combinatorial Bethe ansätze both at q = 1 [15] and q = 0 [19]. These features have been fully worked out in [24] for A (1) 1 . More recently it has been shown further that the set of angle variables can be decomposed into connected components and every such component is a torus [36]. Here we present a conjectural generalization of these results to A (1) n case. It provides a conceptual explanation of the dynamical period and the state counting formula proposed in [25,26].
Recall that each block (a, i) of a rigged configuration is assigned with the rigging r i,α to α ∈ Z uniquely so that the quasi-periodicity i,α ) α∈Z will be called the quasi-periodic extension of (r where denotes a direct product of sets. Using Theorem 2.2, we define an injection where the bottom right is the quasi-periodic extension of the original rigging r = r (a) i,α (aiα)∈H in (µ, r) as explained above. Elements ofJ (µ) will be called extended rigged configurations.
One may still view an extended rigged configuration as the right diagram in (2.5). The only difference from the original rigged configuration is that the riggings r i,α (ai)∈H ,α∈Z ∈J (µ) will be denoted by the same symbol r.
Let T be the abelian group generated by all the time evolutions T (a) l with 1 ≤ a ≤ n and l ≥ 1. Let further A be a free abelian group generated by the symbols s (a) i with (a, i) ∈ H. We consider their commutative actions onJ (µ) as follows: The generator s which is the set of all A-orbits, whose elements are written as A · r with r ∈J (µ). Elements of J (µ) will be called angle variables. See also (4.17). Since T and A act onJ (µ) commutatively, there is a natural action of T on J (µ). For t ∈ T and y = A · r ∈ J (µ), the action is given by t · y = A · (t · r). In short, the time evolution of extended rigged configurations (3.3) naturally induces the time evolution of angle variables.

Linearization of time evolution
Given the level set P(µ) with µ satisfying the condition (2.14), we are going to introduce the bijection Φ to the set of angle variables J (µ). Recall that Σ(p) denotes the connected component of P(µ) that involves the path p.
This implies that any path p can be expressed in the form p = t · p + by some highest path p + ∈ P + (µ) and time evolution t = (T 8 Although such an expression is not unique in general, one can go from P(µ) to J (µ) along the following scheme: Note that the abelian group T of time evolutions acts on paths by (2.4) and on extended rigged configurations by (3.3). The composition (3.6) serves as a definition of a map Φ only if the non-uniqueness of the decomposition p → (t, p + ) is 'canceled' by regarding t · r mod A.
Our main conjecture in this subsection is the following.
Conjecture 3.2. Suppose the condition (2.14) is satisfied. Then the map Φ (3.6) is well-defined, bijective and commutative with the action of T . Namely, the following diagram is commutative This is consistent with the obvious periodicity (T  [3,4]. The map Φ whose essential ingredient is the KKR bijection φ plays the role of the Abel-Jacobi map. Conceptually, the solution of the initial value problem is simply stated as t N (p) = Φ −1 • t N • Φ(p) for any t ∈ T , where t in the right hand side is linear. Practical calculations can be found in Examples 3.2 and 3.5. In Section 4, we shall present an explicit formula for the tropical Jacobi-inversion Φ −1 in terms of a tropical Riemann theta function. The result of this sort was first formulated and proved in [24,20,21] for n = 1.
Example 3.1. Take an evolvable path whose energies are given as From this and (2.13) we find p ∈ P(µ) with µ = ((33222), (41)). The cyclic shift (T 1 ) j (p) is not highest for any j. However it can be made into a highest path p + and transformed to a rigged configuration as follows: Thus action variable (µ (1) , µ (2) ) indeed coincides with µ = ((33222), (41)) obtained from the energy in agreement with Theorem 2.2. Although it is not necessary, we have exhibited the vacancy numbers on the left of µ (1) and µ (2) for convenience. We list the relevant data: The index set H (2.6) corresponding to µ = (µ (1) , µ (2) ) in ( is the order of symmetry which will be explained in Section 3.3. The matrix F (2.8) reads We understand that the riggings here are parts of extended one obeying (3.1). Then from (3.9) and (3.3), the extended rigged configuration for p (3.8) is the left hand side of Here the transformation to the right hand side demonstrates an example of identification by A. These two objects are examples of representative elements of the angle variable for p (3.8). We will calculate (T (a) l ) 1000 (p) in Example 3.5 using the reduced angle variable that will be introduced in the next subsection.

Example 3.2. Consider the evolvable path
which is same as in Example 1.1. The cyclic shift (T 1 ) j (p) is not highest for any j. However it can be made into a highest path p + and transformed to a rigged configuration as Thus p (3.13) belongs to the same level set P(µ) with µ = ((33222), (41)) as Example 3.1, hence the matrix F (2.8) is again given by (3.11). However, it has a different (trivial) order of symmetry γ (a) i given in the following table: Regard (3.14) as a part of extended rigged configuration as in Example 3. To illustrate the solution of the initial value problem along the inverse scheme (3.7), we derive where the equivalence is seen by using s

Decomposition into connected components
As remarked under Conjecture 2.1, the level set P(µ) is a disjoint union of several connected components Σ(p) under the time evolution T . It is natural to decompose the commutative diagram (3.7) further into those connected components, i.e., T -orbits, and seek the counterpart of Σ(p) in J (µ). To do this one needs a precise description of the internal symmetry of angle variables reflecting a certain commensurability of soliton configuration in a path. In addition, it is necessary to separate the angle variables into two parts, one recording the internal symmetry (denoted by λ) 9 and the other accounting for the straight motions (denoted by ω). We call the latter part reduced angle variables. The point is that action of T becomes transitive by switching to the reduced angle variables from angle variables thereby allowing us to describe the connected components and their multiplicity explicitly. These results are generalizations of the n = 1 case [36]. Let m ≥ 1 and p ≥ 0. 10 We introduce the following set by imposing just a simple extra condition onΛ(m, p) (3.2): and consider the splitting of extended rigged configurations into X 1 and X 2 as follows: where ω = ω j,1+δ ij δ ab (bj)∈H,α∈Z .

(3.21)
In particular, T acts on X 1 transitively leaving X 2 unchanged.
From (3.5) and (3.19), the angle variables may also be viewed as elements of X/A. It is the set of all A-orbits, whose elements are written as A · x with x ∈ X. Since T and A act on X commutatively, there is a natural action of T on X/A. For t ∈ T and y = A·x ∈ X/A, the action is given by t · y = A · (t · x). For any representative element (ω, λ) ∈ X/A of an angle variable, we call the X 1 part ω a reduced angle variable. Under the simple bijective correspondence (3.19), the map (3.6) is rephrased as where we have used the same symbol Φ. The conjectural commutative diagram (3.7) becomes Now we are ready to describe the internal symmetry of angle variables and the resulting decomposition. We introduce a refinement of Λ(m, p) (3.17) as follows: where γ is a (not necessarily largest) common divisor of m and p. In other words Λ γ (m, p) is the set of all arrays (λ α ) α∈Z ∈ Λ(m, p) that satisfy the reduced quasi-periodicity λ α+m/γ = λ α + p/γ but do not satisfy the same relation when γ is replaced by a larger γ ′ . Such γ is called the order of symmetry. By the definition one has the disjoint union decomposition: where γ extends over all the common divisors of m and p. Taking the cardinality of this relation using (3.18) amounts to the identity i . We introduce the g × g matrix F γ and the subsets X 2 γ ⊂ X 2 and X γ ⊂ X by where F ai,bj is specified in (2.8). The matrix F hence F γ are positive definite if p   An important property of the subset X γ ⊂ X is that it is still invariant under the actions of both T and A. Let X γ /A be the set of all A-orbits. According to X = γ X γ , we have the disjoint union decomposition: which induces the disjoint union decomposition of the level set P(µ) according to (3.22). Writing the pre-image of X γ /A as P γ (µ), one has P(µ) = ⊔ γ P γ (µ) and the conjectural commutative diagram (3.22) splits into P γ (µ) is the subset of the level set P(µ) characterized by the order of symmetry γ.
We are yet to decompose it or equivalently X γ /A further into T -orbits. Note that one can think of an A-orbit A · x either as an element of X γ /A or as a subset of X γ . Similarly a T -orbit T · (A · x) in X γ /A can either be regarded as an element of (X γ /A)/T or as a subset of X γ /A. We adopt the latter interpretation. Then as we will see shortly in Proposition 3.1, the T -orbits T · (A · x) of X γ /A can be described explicitly in terms of Z g /F γ Z g , the set of integer points on the torus equipped with the following action of T : where h (r) l is specified in (3.20). In what follows, we shall refer to Z g /F γ Z g simply as torus. Given any x = (ω, λ) ∈ X γ , consider the map (3.31) 36]). The map χ is well-defined, bijective and the following diagram is commutative: Combining Proposition 3.1 with Conjecture 3.2 or its refined form (3.29), we obtain an explicit description of each connected component (T -orbit) as a torus. Conjecture 3.3. For any path p ∈ P γ (µ), the map Φ χ := χ • Φ gives a bijection between the connected component Σ(p) and the torus Z g /F γ Z g making the following diagram commutative: The actions of T in Proposition 3.1 and Conjecture 3.3 are both transitive. Conjecture 3.3 is a principal claim in this paper. It says that the reduced angle variables live in the torus Z g /F γ Z g , where time evolutions of paths become straight motions. See Conjecture 3.6 for an analogous claim in a more general case than (2.14).

Remark 3.2.
Due to the transitivity of T -action, the commutative diagram (3.32) persists even if Φ χ is redefined as Φ χ + c with any constant vector c ∈ Z g , meaning that the choice of the inverse image of 0 ∈ Z g /F γ Z g is at one's disposal. With this option in mind, an explicit way to construct the bijection Φ χ is as follows. Fix an arbitrary highest path p + in Σ(p) and set Φ χ (p + ) = 0. For any p ′ ∈ Σ(p) let t be an element of T such that p ′ = t · p + . One can always find such t since T acts on Σ(p) transitively. If t is written as t = r,l T (r) l d (r) l , then we set i+1 + 1 (i < g a ), = 1 (i = g a ). Let T be the free abelian group generated by all T  In Example 3.3, we have seen that the reduced angle variable of p is given by where F γ is specified in (3.27). The velocity vector h  Next we consider T (1) 3 1000 (p). The reduced angle variable for this path is This means T 1 + 10h (2) 1 + t (1, 0, 1, 0), l = 3, 21h (1) 1 + 3h to these results, one can check that (3.34) leads to (3.33).

Bethe ansatz formula from size and number of connected components
The results in the previous section provide a beautiful interpretation of the character formula derived from the Bethe ansatz at q = 0 [19] in terms of the size and number of orbits in the periodic A (1) n SCA. Let us first recall the character formula: (3.36) In (3.35), the sum is taken over n-tuple of Young diagrams µ = (µ (1) , . . . , µ (n) ) without any constraint. All the quantities appearing in (3.36) are determined from µ by the left diagram in (2.5) and (2.6)-(2.8). The fact Ω(µ) ∈ Z can be easily seen by expanding det F . It is vital that the binomial coefficient here is the extended one: which can be negative outside the range 0 ≤ b ≤ a. Due to such negative contributions, the infinite sum (3.35) cancels out except leaving the finitely many positive contributions exactly when L ≥ |µ (1) | ≥ · · · ≥ |µ (n) |. i . Under such a string hypothesis, the Bethe equation becomes a linear congruence equation at q = 0 (3.42), and counting its off-diagonal solutions yields Ω(µ). In this sense, the identity (3.35) implies a formal completeness of the Bethe ansatz and string hypothesis at q = 0. For more details, see Section 3.6, especially Theorem 3.3. A parallel story is known also at q = 1 [15] as mentioned under Theorem 2.2.
Back to our A n SCA, it is nothing but the integrable vertex model at q = 0, where U q A (1) n modules and row transfer matrices are effectively replaced by the crystals and time evolution operators, respectively. In view of this, it is natural to link the Bethe ansatz formula Ω(µ) with the notions introduced in the previous subsection like level set, torus, connected components (T -orbits) and so forth. This will be done in this subsection, providing a conceptual explanation of the earlier observations [25,26]. Our main result is stated as Theorem 3.1. Assume the condition (2.14) and Conjecture 3.3. Then the Bethe ansatz formula Ω(µ) (3.36) counts the number of paths in the level set P(µ) as follows: (3.37) Here γ = γ i . The result (3.37) uncovers the SCA meaning of the Bethe ansatz formula (3.36). It consists of the contributions from sectors specified by the order of symmetry γ. Each sector is an assembly of identical tori (connected components), therefore its contribution is factorized into its size and number (multiplicity).
For the proof we prepare a few facts. Note that s (a) i (3.21) sending λ to λ ′ also defines an action of A on X 2 γ part alone.
Lemma 3.1. The number of T -orbits in X γ /A is |X 2 γ /A|, i.e., the number of A-orbits in X 2 γ .
Proof . Let y = A · x and y ′ = A · x ′ be any two elements of X γ /A. They belong to a common T -orbit if and only if there exists t ∈ T such that t · y = y ′ . It is equivalent to saying that there exist t ∈ T and a ∈ A such that t · (a · x) = x ′ . Write x, x ′ as x = (ω, λ) and x ′ = (ω ′ , λ ′ ). Suppose λ and λ ′ belong to a common A-orbit in X 2 γ . Then there exits a ∈ A such that a · λ = λ ′ . Hence we have a · x = (ω, λ ′ ) with someω ∈ X 1 . Since T acts on the X 1 part transitively and leaves the X 2 γ part untouched, there exits t ∈ T such that t · (a · x) = x ′ . Suppose λ and λ ′ belong to different A-orbits. Then a · x and x ′ have different X 2 γ parts for any a ∈ A. Hence for any t ∈ T and a ∈ A, we have t · (a · x) = x ′ .
Thus y and y ′ belong to a common T -orbit if and only if λ and λ ′ belong to a common A-orbit. The proof is completed.
Lemma 3.2. The cardinality of the set X 2 γ /A is given by Proof . It is sufficient to show that each factor in the right hand side gives the number of A-orbits for each (a, i) block. So we omit all the indices below and regard A as a free abelian group generated by a single element s. Let λ = (λ α ) α∈Z be any element of Λ γ (m, p). Then due to (3.21) we have s n · λ = (λ α+n − λ 1+n ) α∈Z . By (3.23) and (3.17) this implies that s m/γ · λ = λ and there is no 0 < k < m/γ such that s k · λ = λ. Hence the number of A-orbits in Λ γ (m, p) is |Λ γ (m, p)|/(m/γ). Proof . This is due to Conjecture 3.3 and det F γ > 0 under the assumption (2.14). See the remark under (3.25). Now we are ready to give Proof of Theorem 3.1.
Accordingly the level set P(µ) is decomposed into 36 tori as The paths (3.8) and (3.13) in Examples 3.1 and 3.2 belong to one of the tori Z 4 /F γ 2 Z 4 and Z 4 /F γ 1 Z 4 , respectively.

Dynamical period
Given a path p ∈ P γ (µ), the smallest positive integer satisfying (T l . Here we derive an explicit formula of the dynamical period as a simple corollary of Conjecture 3.3. It takes the precise account of the symmetry specified by γ and refines the earlier conjectures in [25,26] for n ≥ 2 which was obtained from the Bethe eigenvalues at q = 0.
For nonzero rational numbers r 1 , . . . , r s , we define their least common multiple by Given (r, l) with 1 ≤ r ≤ n and l ≥ 1, define F γ [bj] (resp. F [bj]) to be the g × g matrix obtained from F γ (3.25) (resp. F (2.8)) by replacing its (bj)th column by h (r) l (3.20). Set where the LCM should be taken over only those (bj) satisfying det F [bj] = 0.

Relation to Bethe ansatz at q = 0
Let us quickly recall the relevant results from the Bethe ansatz at q = 0. For the precise definitions and statements, we refer to [19]. Consider the integrable U q A (1) n vertex model on a periodic chain of length L. If the quantum space is the L-fold tensor product of the vector representation, the Bethe equation takes the form: for 1 ≤ a ≤ n and 1 ≤ i ≤ µ (a) . Here L ≥ µ (1) ≥ · · · ≥ µ (n) specifies a sector (2.20) preserved by row transfer matrices. The parameter is related to q by q = e −2π . Fix an n-tuple of Young diagrams, i.e., the string content µ = µ (1) , . . . , µ (n) as in (2.5). We keep the notations (2.6)-(2.9). By string solutions we mean the ones in which the unknowns u iαk stands for a small deviation. u

(a)
i,α is the string center of the αth string of color a and length i. For a generic string solution, the Bethe equation is linearized at q = 0 into a logarithmic form called the string center equation: for (aiα) ∈ H. Here the G × G coefficient matrix A = (A aiα,bjβ ) (aiα),(bjβ)∈H is specified as Last, we prohibit the collision of string centers u for any (a, i) ∈ H. This is a remnant of the well-known constraint on the Bethe roots so that the associated Bethe vector does not vanish. To summarize, we consider off-diagonal solutions u = u  Given an extended rigged configuration r = r (a) i,α (ai)∈H ,α∈Z ∈J (µ), we define a map The proof is parallel with [24,Section 4.2] for the n = 1 case. The induced bijection will also be denoted by Ψ. It also induces the action of the time evolutions T on U (µ) by T Their cardinality is given by with Ω(µ) defined in (3.36). We will argue the time evolutions of the Bethe roots further in Section 4.2.

General case
Let P(µ) be a level set. From µ, specify the data like m  i ≥ 1 (2.14). In this subsection we treat the general configuration, namely we assume (3.46) It turns out that the linearization scheme remains the same provided one discards some time evolutions and restricts the dynamics to a subgroup T ′ of T . We begin by preparing some notations about partitions. Let λ, ν be partitions or equivalently Young diagrams. We define λ ∪ ν to be the partition whose parts are those of λ and ν, arranged in decreasing order [27]. For example, if λ = (4221) and ν = (331), then λ ∪ ν = (4332211).
We denote by 2 Let λ be a partition. We say that λ covers the block 2   does not cause an inadmissible time evolution even if it is null and convex. We will also say that T (a) l is admissible or inadmissible to a path p ∈ P(µ) depending on whether T (a) l is admissible or inadmissible to µ. As a generalization of Conjecture 2.1, we propose   Let T ′ be the abelian group generated by all the time evolutions T (a) l admissible to µ. Then Conjecture 3.4 implies that T ′ acts on the level set P(µ). We define the connected component Σ ′ (p) to be the T ′ -orbit in P(µ) that contains p.
As it turns out, the restriction of the dynamics from T to T ′ will be matched by introducing a sub-lattice L of Z g as follows:   = µ (1) , . . . , µ (n) be a configuration, i.e., (3.46) is satisfied. Suppose the (a, i) block of µ is null and convex. Then: Recall that the matrix F γ is defined by (2.8) and (3.25).
By (i) of Lemma 3.4 we have p (a) 3 = 0, and by (ii) of the same lemma any column ofF a is an integer multiple (and possible integer addition to the 4-th and 5-th rows) of one of the following vectors: where x and y are some integers. Clearly these vectors can be expressed by linear combinations of the columns of the matrices in (3.47) with integer coefficients. Proof . By using the above example we have The general case is similar.
Let the abelian group T ′ act on the torus L/F γ Z g by (3.30). This action is transitive, since T ′ contains the free abelian subgroup generated by all T  Owing to this property, one can introduce Φ ′ by the same scheme as (3.6) with T replaced by T ′ . Similarly, let χ ′ : T ′ · (A · x) −→ L/F γ Z g be defined as in (3.31). As the generalization of Conjecture 3.3, we have Conjecture 3.6. For any path p ∈ P γ (µ), the map Φ ′ χ := χ ′ • Φ ′ gives a bijection between the connected component Σ ′ (p) and the torus L/F γ Z g making the following diagram commutative: , where the product is taken over all null and convex blocks 2 (a) i . Note that this factor is common to all the connected components in the level set. Thus Conjecture 3.6 also tells that , which refines [26,Conjecture 4.2]. l 's are admissible. The matrix F γ for p III is given by The cardinality of the connected component in the level set is calculated as  The matrix F γ for p IV is given by The cardinality of the connected component is calculated as = 2048/(2 · 2) = 512.

Summary of conjectures
Here is a summary of our conjectures. Each one is based on those in the preceding lines. Thus the principal ones are the linearizations. Let µ (a) , r (a) be the color a part of the rigged configuration (µ, r) depicted in the right diagram of (2.5). We keep the notations in (2.6)-(2.9). An explicit formula for the image path of the KKR map φ −1 (2.18) is known in terms of the tropical tau function [23]. It is related to the charge function on rigged configurations and is actually obtained from the tau function in the KP hierarchy [11] by the ultradiscretization with an elaborate adjustment of parameters from the KP and rigged configurations: Here the sums range over (aiα) ∈ H wherever N (a) i,α is involved. Thus in the second line of (4.1), j,β is to be understood as 0 when d = n + 1. In (4.1), min is taken over N (a) i,α ∈ {0, 1} for all (aiα) ∈ H. Thus it consists of 2 G candidates. Obviously, τ k,d ∈ Z ≥0 holds.
Theorem 4.1 ([23]). The image of the KKR map φ −1 is expressed as follows: (2 ≤ a ≤ n + 1). (4.2) In the context of the box-ball system, x k,a represents the number of balls of color a in the kth box from the left for 2 ≤ a ≤ n + 1. The remaining x k,1 is determined from this by x k,1 = 1 − x k,2 − · · · − x k,n+1 , which also takes values x k,1 = 0, 1 in the present case.
In the lemma, ⊔ stands for the sum (union) as the multisets of strings, namely, the rows of µ attached with riggings. More formally a string is a triple a, l i,α that is labeled with H. Let p ∈ B 1,1 ⊗L be the highest path corresponding to the rigged configuration (µ, r). From Lemma 4.1, the rigged configuration of the highest path p ⊗M ∈ B 1,1 ⊗M L is µ, r 1 ⊔ µ, r 2 ⊔ · · · ⊔ µ, r M , where r k = r to the riggings in the (a, i) block for each (a, i). This reminds us of (3.1), and is in fact the origin of the extended rigged configuration in Section 3.1.
We proceed to the calculation of the tropical tau function τ M k,d associated with the above rigged configuration µ, r 1 ⊔ µ, r 2 ⊔ · · · ⊔ µ, r M . In (4.1), the variable N where the minimum is now taken over n (a) i,α ∈ {0, 1, . . . , M − 1} for all the blocks (a, i). The notation can be eased considerably by introducing a quadratic form of n = n (a) i,α (aiα)∈H as follows: here should be distinguished from the g-dimensional one h We call B (4.6) the tropical period matrix although a connection to a tropical curve is yet to be clarified. We also introduce the G-dimensional vector 1 = (1) (aiα)∈H . Then from (2.7) and (4.8) we get (4.11) The velocity vector h Now we take M to be even and make the shifts n → n + M 2 1 and k → k ′ = k + M L 2 . Using (4.9), we find The first term is formally identical with (4.5) but now the minimum extends over − M 2 ≤n (a) i,α < M 2 . The scalars u M , v M , w M are independent of k ′ and d, therefore these terms are irrelevant when Here Θ denotes the tropical Riemann theta function which enjoys the quasi-periodicity In the context of the periodic box-ball system, the tropical Riemann theta function was firstly obtained in this way in [20] for rank n = 1 case. See [9,10,30] for an account from the tropical geometry point of view. Another remark is that Proposition 4.1 and (4.13) directly lead to the tropical Hirota equation for our Θ: 1 . For n = 1, see also [10]. From (4.2), we arrive at our main formula in this section.
Note that this reduces a solution of the initial value problem p → T (r 1 ) l 1 · · · T (r N ) l N (p) to a simple substitution r → r + h (r 1 ) Here S m denotes the symmetric group of degree m. In fact the set J (µ) of angle variables is naturally described as To go to the right hand side, one just forgets the inequality r i,α+1 within each block (ai) ∈ H identifying all the re-orderings. Compare also the matrix B (4.6) with (3.4).
When n = 1, one can remove the assumption 'Under Conjecture 3.2' in Theorem 4.3 due to [24]. If further ∀ m What makes the calculation tedious after this is that one has to classify n i . This was done in [21] for n = 1. Here we omit the detail and only mention that the result is expressed in terms of a rational characteristic tropical Riemann theta function with the g × g reduced period matrix: Let us proceed to an explicit Θ formula for a carrier of type B 1,l . Consider the highest path p = x 1 ⊗ x 2 ⊗ · · · ⊗ x L ∈ P + (µ) in Theorem 4.2 expressed as (4.16). We consider the calculation of the time evolution T (1) (2.2). Locally it is depicted as From the proof of Lemma D.1, a carrier satisfying the periodic boundary condition y 0 = y L ∈ B 1,l and thereby inducing the time evolution T (1) l can be constructed by u 1,l ⊗ p ≃p ⊗ y 0 . Here u 1,l ∈ B 1,l is the semistandard tableau of length l row shape whose entries are all 1. This fixes the carriers in the intermediate stage y 1 , y 2 , . . . , y L−1 ∈ B 1,l by y 0 ⊗(x 1 ⊗· · ·⊗x k ) These shortcoming are fixed of course by refining the construction of joint eigenvectors along the connected components. Instead of trying to split the sum (4.20) into them, we simply introduce for each angle variable (ω, λ) ∈ X γ /A having the order of symmetry γ.
Here v(φ, λ) ∈ P γ (µ) is the path corresponding to (φ, λ) ∈ X γ /A, and d λ ∈ Z g can be chosen arbitrarily. Then by noting T (r) This eigenvalue is indeed an N (r) l th root of unity due to the proof of Theorem 3.2. Given λ, there are det F γ independent vectors |ω, λ (4.25). On the other hand, the number of the choices of λ ∈ X 2 γ /A is given by Lemma 3.2. Obviously these vectors are all independent because the set of monomials involved in |ω, λ and |ω ′ , λ ′ are the set of paths that are disjoint if λ = λ ′ . From this fact and (3.37), we obtain the refinement of (4.23) according to the order of symmetry γ and further (bit tautologically) according to the connected components: where for example p 0 = Φ −1 ((ω = 0, λ)) ∈ P γ (µ).

Miscellaneous calculation of time average
As a modest application of the formulas by the tropical Riemann theta function, we first illustrate a calculation of some time average along the periodic box-ball system (n = 1). Analogous results will be stated for general n in the end. We use the terminology in the periodic box-ball system.
Let p ∈ (B 1,1 ) ⊗L be the path with the angle variable I. The time evolution T (1) l will simply be denoted by T l . Set T t l (p) = x 1 (t)⊗· · ·⊗x L (t). For n = 1, one can label x k (t) = (x k,1 (t), x k,2 (t)) ∈ B 1,1 just by x k,2 (t), which from now on will simply be denoted by x k (t)(= 0, 1). It represents the number of balls in the kth box. Then (4.16) reads The notations (4.6)-(4.8) are simplified hereafter as Here l i , m i are the shorthand of l i that specify the action variable (single partition) µ = µ (1) as in (2.5). The relation (4.9) reads (4.28) Note that the total number of balls at any time is |µ| = g i=1 l i m i . This also follows immediately from (4.26) as where we have used (4.28) and the quasi-periodicity (4.15). We introduce the density of balls: Let N l be the dynamical period under T l . Thus x k (N l ) = x k (0) holds for any 1 ≤ k ≤ L.
Proposition 4.2. The time average of x k (t) under T ∞ over the period N ∞ is given by Proof . Using (4.26) with l = ∞, we find that N∞−1 t=0 x k (t) is equal to Note that the assumption implies N ∞ h ∞ ∈ BZ G . Thus one can reduce this by applying the quasi-periodicity (4.15), obtaining Actually (4.29) follows at once without this sort of calculation if the left hand side is assumed to be independent of k. However, such homogeneity under spatial translation is not always valid for some time evolution T l with l < ∞ and the initial condition that possess a special commensurability. Denote the time average 1 N l N l −1 t=0 Q(t) of a quantity Q(t) under T l by Q l . Then a trivial corollary of Proposition 4.2 is In particular, this average vanishes if ω is a nontrivial Lth root of unity.
Let us proceed to a less trivial example. In [22], the number of balls in the carrier for any time evolution T l was expressed also in terms of the tropical Riemann theta function. The carrier that induces the time evolution p → T l (p) is the element v l ∈ B 1,l such that v l ⊗ p ≃ T l (p) ⊗ v l under the isomorphism of crystals B 1,l ⊗ B 1,1 ⊗L ≃ B 1,1 ⊗L ⊗B 1,l . See the explanation around (2.2). There uniquely exists such v l for any p with density ρ < 1/2 as shown in [24, Proposition 2.1]. Let us consider the intermediate stage of sending the carrier y 0 (t) := v l to the right by repeated applications of combinatorial R: v l ⊗ x 1 (t) ⊗ · · · ⊗ x k (t) ⊗ x k+1 (t) · · · ⊗ x L (t) ≃ x 1 (t + 1) ⊗ · · · ⊗ x k (t + 1) ⊗ y k (t) ⊗ x k+1 (t) · · · ⊗ x L (t).
The preceding result (4.18) reduces to this upon setting n = 1, a = 2 and r − p 2 = J + th l . Proposition 4.3. The time average of the number of balls in the carrier with capacity l at position k under T l is given by Proof . The proof is parallel with the one for Proposition 4.2 by using N l h l ∈ BZ G and the quasi-periodicity (4.15).
Note that the result is independent of the order of symmetry γ as well as k. We list average y k l with the dynamical period N l .   (2 ≤ a ≤ n + 1). for n = 1 due to h (n+1) l = 0. See (4.8). In particular at l = 1 we have y k,a 1 = 1 L t 1 h (a−1) ∞ − h (a) ∞ = |µ (a−1) | − |µ (a) | L by means of (4.9) and (4.10). From (2.20), this is the density of the color a balls in the path B 1,1 ⊗L , which is natural in view of (2.3). The average of the total number of balls is n+1 a=2 y k,a l = t h (1) ∞ , which again reduces to Proposition 4.3 for n = 1.

A Row and column insertions
Let T be a semistandard tableau. The row insertion of the number r into T is denoted by T ← r and defined recursively as follows: Here we have denoted the entries in the first row of T by i 1 ≤ i 2 ≤ · · · ≤ i l , and the other part of T by T . The finished case includes the situation T = ∅. In the to be continued case, r bumps out the smallest number i k that is larger than r (row bumping). The tableau T ← r is obtained by repeating the row bumping until finished. The column insertion of the number r into T is denoted by r → T and defined recursively as follows: Here we have denoted the entries of the first column of T by i 1 < i 2 < · · · < i l , and the other part of T by T . The finished case includes the situation T = ∅. In the to be continued case, r bumps out the smallest number i k that is not less than r (column bumping). The tableau r → T is obtained by repeating the column bumping until finished.
It is straightforward to check that the gauge transformed simple reflection K a is equal to K a defined by the procedure (i)-(iv) explained before Theorem 2.1 by using the signature rule explained in [24,Section 2.3].

C.1 General remarks
The original KKR bijection [15,16] is the one between rigged configurations and Littlewood-Richardson tableaux. Its ultimate generalization in type A (1) n corresponding to B r 1 ,l 1 ⊗· · ·⊗B r 1 ,l L is available in [18]. In the simple setting of this paper, the Littlewood-Richardson tableaux are in one to one correspondence with highest paths in B = (B 1,1 ) ⊗L . The KKR bijection in this paper means the one (2.18) between rigged configurations and those highest paths. See [23,Appendix C] for an exposition in a slightly more general setting B 1,l 1 ⊗ · · · ⊗ B 1,l L .
Recall that a rigged configuration (µ, r) is a multiset of strings, where a string is a triple a, l We also recall that one actually has to attach the data L with (µ, r) to specify the vacancy numbers p (a) i (2.7). Thus we write a rigged configuration as (µ, r) L . We regard a highest path b 1 ⊗ · · · ⊗ b L ∈ (B 1,1 ) ⊗L as a word b 1 b 2 . . . b L ∈ {1, 2, . . . , n + 1} L . The algorithms explained below obviously satisfy the property (2.20).