Symmetry, Integrability and Geometry: Methods and Applications Level Set Structure of an Integrable Cellular Automaton ⋆

Based on a group theoretical setting a sort of discrete dynamical system is constructed and applied to a combinatorial dynamical system defined on the set of certain Bethe ansatz related objects known as the rigged configurations. This system is then used to study a one-dimensional periodic cellular automaton related to discrete Toda lattice. It is shown for the first time that the level set of this cellular automaton is decomposed into connected components and every such component is a torus.


Introduction
The Liouville's theorem on completely integrable systems is one of the most fundamental results in classical mechanics. By V.I. Arnold's formulation one of the claims in the theorem says that any compact connected level set of completely integrable systems with N degrees of freedom is diffeomorphic to an N -dimensional torus [1]. It is called an invariant torus and the theorem also claims that the phase flow with the Hamiltonian function determines a conditionally periodic motion on it.
In this paper we study the level set structure of a one-dimensional cellular automaton known as the periodic box-ball system (pBBS) [2,3] and construct its invariant tori. It is one of the (ultra-)discrete dynamical systems associated with integrable non-linear evolution equations. See the "picture" in Subsection 4.1, just below Remark 4, for an example of the time evolution of this system, in which mutually interacting solitons are traveling along it. This system is attracting attentions because of its relations with discrete Toda lattice [4], Bethe ansatz of integrable quantum spin chains [5], tropical geometry [6] and ultradiscrete Riemann theta functions [7].
The purpose of this paper is twofold. The first is to construct a discrete dynamical system through a group theoretical setting which has potentially several applications in (ultra-)discrete integrable systems (Theorem 1). The second is to make the structure of the level set of pBBS perfectly clear as one of its applications (Theorem 2). It is shown for the first time that the level set is decomposed into connected components and every such component is a torus, as if it were a compact level set of completely integrable systems in Hamiltonian mechanics.
Let us begin by describing backgrounds and motivations on the problem more precisely. Consider the completely integrable systems with N degrees of freedom again. Such a system has N independent first integrals. Its level set is determined by fixing the values of the first integrals, and becomes an N -dimensional manifold in the 2N -dimensional phase space. Inspired by this picture in Hamiltonian mechanics, A. Kuniba, A. Takenouchi and the author introduced the notion of the level set of pBBS [5], and provided its volume formula. In fact it was exactly equal to the same formula for the enumeration of the off-diagonal solutions to the string center equations in combinatorial Bethe ansatz [8]. It is based on the notion of rigged configurations [9,10], and is described in the following way. Let m j be the number of solitons with amplitude j. In what follows m j are supposed to be positive for 1 ≤ j ≤ N and to be zero for j > N for simplicity. One can think of m = (m j ) 1≤j≤N as a collection of fixed values of the first integrals. We denote by J (m) the level set specified by m and by Ω(m) its volume. Then we have [5] Ω(m) = (det F ) Here p j are positive integers and F is an N × N matrix with integer entries, which are explicitly expressed in terms of m and the system size L.
It can be shown that if there is no solitons of common amplitudes then the level set J (m) becomes a torus F Z N \Z N . 1 This fact is suggested by the volume formula (1) that reduces to Ω(m) = det F when m j = 1 for every j. However this simple picture fails when there are multiple solitons of common amplitudes. If so, the right hand side of (1) can not be regarded as the volume of a torus any more. An expression for the level set in this generalized case is given by where I n = S n \(Z n − ∆ n ) is the n dimensional lattice without the diagonal points ∆ n = {(z 1 , . . . , z n ) ∈ Z n | z α = z β for some 1 ≤ α = β ≤ n} and identified under the permutations S n . The A is a γ × γ symmetric matrix (γ = m 1 + · · · + m N ) with integer entries [5].
In this paper we show that the level set (2) has generally many connected components, and every such component is written as F (α) Z N \Z N where F (α) is an N × N matrix determined by the above F and the symmetry α of the system that depends on the initial conditions. It is also shown that the time evolutions of the cellular automaton yield straight motions on the torus, just like the phase flows on the invariant torus generated by first integrals.
The layout of the rest of this paper is as follows. In Section 2 we begin with an abstract group theoretical setting and present a general result, and one of its specializations called a direct product model. Here we establish our first main result, Theorem 1. This model is interpreted as a discrete dynamical system. In Section 3 we construct a specific example of the direct product model associated with the rigged configurations. We give a review on the periodic boxball system in Section 4, and then construct its invariant tori based on the results in previous sections. Our second main result is Theorem 2. Properties of the cellular automaton that follows from this result are discussed in Section 5. Two elementary lemmas are given in Appendix A, and an algorithm for calculating rigged configurations is presented in Appendix B.
2 Construction of a discrete dynamical system 2.1 Group theoretical setting Let X be a set, S(X) be the group of all bijections from X to itself. Then S(X) acts on X from the left by σ · x = σ(x) (∀ x ∈ X, ∀ σ ∈ S(X)). From now on we assume that every group action is left action and omit the word "from the left". Given any group G acting on X there is an equivalence relation associated with its action. Its equivalence classes are called G-orbits on X. If there is only one G-orbit, the action of G is called transitive.
The G-orbit containing x ∈ X is denoted G · x, and is called the G-orbit of x. The set of all G-orbits on X is denoted G\X. Though one can think of G · x either as an element of G\X or as a subset of X, we shall adopt the latter interpretation and elements of G\X will be written There is a transitive action of G on G · x(⊂ X). The following lemma is elementary. Lemma 1. Let G and H be any groups that act on X commutatively. Then G naturally acts on H\X. Namely their is a unique action of G on H\X that is commutative with the canonical Proof . Given any p ∈ H\X one can write it as p = [x] H for some x ∈ X. We define Γ : G × H\X → H\X to be a map given by the relation Γ (g, p) = [g · x] H for all g ∈ G. It is easy to see that this map is well-defined and yields the desired action of G by g · p = Γ (g, p).
Let G 1 and G 2 be subgroups of S(X) that act on X commutatively. Then G 1 ∩ G 2 is also a subgroup of S(X) and its action on X is commutative with those of G 1 and G 2 . By Lemma 1 there is a natural action of G 1 on G 2 \X. Let x be any element of X. Then G 1 · [x] G 2 is a subset of G 2 \X where G 1 acts on transitively. Similarly G 1 · x is a subset of X where G 1 and G 1 ∩ G 2 act on commutatively. Hence by Lemma 1 there is a natural action of G 1 on (G 1 ∩ G 2 )\(G 1 · x). Proposition 1. Let G 1 and G 2 be subgroups of S(X) and suppose their actions on X is com- Well-definedness: Suppose there is another g ′ ∈ G 1 such that z = g ′ · [x] G 2 . Since the actions of G 1 and G 2 are commutative we have Hence one can take h as an element of Commutativity with G 1 action: Surjectivity: Choose an arbitrary element C of (G 1 ∩G 2 )\(G 1 ·x). Then there exists y ∈ G 1 ·x Among the assumptions of Proposition 1, the condition (G 1 · x) ∩ (G 2 · x) ⊂ (G 1 ∩ G 2 ) · x is rather specific. In the next subsection we construct an example of the triplet (G 1 , G 2 , X) that satisfies this condition for all x ∈ X.

Direct product model
Let X 1 , X 2 be sets and H 1 , H 2 groups that act on X 1 , X 2 respectively. Choose an arbitrary subgroup H ′ 2 of H 2 . We define In other words X ′ 2 is the set of all elements of X 2 whose stabilizer associated with the H 2 action is H ′ 2 . In what follows H 2 is supposed to be abelian, and we denote the multiplication as a sum g + h ∈ H 2 for any g, h ∈ H 2 . Then X ′ 2 is invariant under the action of H 2 . Remark 1. Suppose there are several different actions of the group G on the set X. To distinguish them we write Γ : G × X → X for example to denote a specific action of G on X. For any such Γ there is a group homomorphism ρ Γ : G → S(X) such that the relation Γ (g, x) = ρ Γ (g)·x holds for all g ∈ G, x ∈ X. Such group homomorphisms are called permutation representations.
We define Γ 1 : H 1 × X → X to be a diagonal action of H 1 on X in which its action on X ′ 2 is trivial. In other words we have Suppose there exists a map Γ 2 : H 2 × X → X that has the following property. For all Then Γ 2 yields an action of H 2 on X.
By definition we have Γ 2 (g, x) = (ϕ(g, yields an action of H ′ 2 on X in which its action on X ′ 2 is trivial.
where ρ Γ 1 and ρ Γ 2 are the permutation representations. Note that G 1 and G 2 depend on the choice of H ′ 2 . In what follows the action of H 1 on X 1 is supposed to be transitive.
on X ′ 2 is trivial, and the action of G 1 on X 1 is transitive.

Lemma 3. The following relation holds
for all x ∈ X = X 1 × X ′ 2 . Proof . By Lemma 2 we have for all x ∈ X, where the latter inclusion is obvious. The opposite inclusions are proved as follows.
Take any x = (x 1 , x 2 ) ∈ X. Choose an arbitrary element y of (G 1 · x) ∩ (G 2 · x). Since y lies in ). Then since y lies in G 1 · x this implies g · x 2 = x 2 , forcing g to be an element of H ′ 2 by (3). Hence y is an element of In what follows the actions of G 1 and G 2 on X are supposed to be commutative. By Proposition 1 and Lemma 3 there is a bijection between G 1 · [x] G 2 and G 1 ∩ G 2 \(G 1 · x) that is commutative with the action of G 1 = ρ Γ 1 (H 1 ). Note that Lemma 3 implies G ′ 2 = G 1 ∩G 2 . Note also that for any x = (x 1 , x 2 ) ∈ X we have G 1 · x = X 1 × {x 2 } ∼ = X 1 as a subset of X, for the action of G 1 on X 1 is transitive and that on X 2 is trivial. Thus we have the following. Theorem 1. Let the sets X 1 , X ′ 2 , the groups G 1 , G 2 , G ′ 2 , and their actions on the set X = X 1 × X ′ 2 be defined as above, and x be any element of X. Then there is a bijection between G 1 · [x] G 2 and G ′ 2 \X 1 that is commutative with the action of G 1 . This is our first main result in this paper.

Interpretation as a dynamical system
In this subsection we present, without mathematical rigor, an interpretation of the direct product model as a dynamical system. It is intended to help readers to have some intuitive physical images on the model. Consider any completely integrable system with N degrees of freedom in Hamiltonian mechanics. There are N independent first integrals. By fixing their values we obtain the level set, an N -dimensional manifold in the phase space R 2N . On the level set there are N mutually commuting phase flows associated with the first integrals. Note that the time evolution of the system is one of the phase flows since the Hamiltonian itself is one of the first integrals.
One can think of the triplet (G 1 , G 2 , X) in Subsection 2.2 as a discrete analogue of such a completely integrable system. LetX := X 1 × X 2 be a non-compact level set and suppose the groups G 1 , G 2 are acting onX commutatively. We regard the group action of G 2 as the symmetry of the system and assume that it yields a compact (but not necessarily connected) level set G 2 \X. Then each state of the system is represented by a G 2 -orbit [x] G 2 for some x ∈X. We regard G 1 as the group generated by the mutually commuting phase flows associated with the first integrals. Then one can think of Suppose the group G 2 acts not only onX but also on X 2 . Choose an arbitrary subgroup G ′ 2 of G 2 and denote by X ′ 2 the set of all elements of X 2 whose stabilizer associated with the G 2 action is G ′ 2 . Let x be any element of X = X 1 × X ′ 2 . Now the model in Subsection 2.2 can be interpreted as follows. There is a bijection between the connected component of the level set G 1 · [x] G 2 and the quotient set G ′ 2 \X 1 that is commutative with the phase flows of the system. In particular the time evolution of the system is commutative with this bijection.
Consider the completely integrable system in Hamiltonian mechanics again. Suppose the level set is compact and connected. Then by Arnold-Liouville theorem it must be diffeomorphic to an N -dimensional torus [1]. It is called an invariant torus. In the following sections we construct a direct product model with X 1 = Z N and G ′ 2 being a sub-lattice of Z N which acts on X 1 by left translation. Now the quotient set G ′ 2 \X 1 is, roughly speaking, a discrete analogue of invariant torus.

Extended rigged configurations
The rigged configuration is an ingenious device utilized in combinatorial Bethe ansatz [9,10]. We construct an example of the direct product model associated with the rigged configurations. The result of this section will be used in Subsection 4.4.
We introduce a pair of sets X 1 and X 2 . Choose an arbitrary positive integer N and define X 1 = Z N . For any pair of positive integers m, p we define Given any positive integer sequences (m j ) 1≤j≤N , (p j ) 1≤j≤N we define X 2 = N j=1 Λ(m j , p j ). Each element of X 2 is written, for instance, as λ = (λ (j) ) 1≤j≤N with λ (j) ∈ Λ(m j , p j ) or Example 1. Let (m 1 , m 2 , m 3 ) = (3, 2, 1) and (p 1 , p 2 , p 3 ) = (12,6,4). Each element of the set X 2 = Λ(3, 12) × Λ(2, 6) × Λ(1, 4) is written as λ = (λ (1) , λ (2) , λ (3) ). It is labeled by three integers a = λ 3 and c = λ The setX = X 1 × X 2 with the above X 1 , X 2 is regarded as the non-compact level set considered in Subsection 2.3. It will be identified with the set of extended rigged configurations in Subsection 4.3 under a certain condition imposed on the values of m j and p j . For the time being we ignore this condition and callX itself the set of extended rigged configurations.
We introduce a pair of abelian groups H 1 and H 2 . Suppose H 1 acts on X 1 by where ψ Γ : H 1 → X 1 is a map common to every ω ∈ X 1 . This map must be linear, i.e. for all . We assume that this action is transitive, i.e. for any ω, ω ′ ∈ X 1 there exists g ∈ H 1 such that Γ (g, ω) = ω ′ . An example of such a map will appear in the next section, just above Lemma 6.
Remark 2. Since X 1 = Z N , any sub-lattice of X 1 can be thought of as an abelian group which acts on X 1 by translation. As a sub-lattice of X 1 we can take X 1 itself, which leads to the above H 1 and its action on X 1 .
Suppose H 2 is a free abelian group with generators s 1 , . . . , s N . We define Υ : 3 and c = λ Here a, b, c are the labels for λ in Example 1.

Cyclic group structures
Let ρ Υ : H 2 → S(X 2 ) be the permutation representation for the action of Υ in the previous subsection. Then ρ Υ (H 2 ) is isomorphic to C m 1 × · · · × C m N , where C m is an order m cyclic group. This in particular determines an action of C m on Λ(m, p). The following facts are well known in the theory of cyclic groups.

Proposition 2. Every subgroup of a cyclic group is a cyclic group.
Proposition 3. Let m be a positive integer and n be a divisor of m. Then there exists a unique subgroup of C m that is isomorphic to C n .
Let α be a common divisor of m and p. As a subset of Λ(m, p) we define In other words Λ (α) (m, p) is the set of all elements of Λ(m, p) whose stabilizer associated with the action of C m is isomorphic to C α . Now we have the decomposition Λ(m, p) = α Λ (α) (m, p) where α runs over every common divisor of m and p. Choose a sequence α = (α j ) 1≤j≤N with each α j being a common divisor of m j and p j . Let H is the set of all elements of X 2 whose stabilizer associated with the H 2 action is H Example 3. Consider the X 2 in Example 1. We have Λ (3,12) The X (α) 2 here corresponds to the X ′ 2 , and the X (α) to the X in Subsection 2.2. The set X = X 1 × X 2 is decomposed asX = α X (α) where each α j runs over every common divisor of m j and p j .

Group actions on the set of rigged configurations
We define a pair of commutative actions of H 1 and H 2 onX = X 1 × X 2 . As in Subsection 2.2, let Γ 1 : Fix an arbitrary set of In the next section we take B j,k to be those given in (12). For each g ∈ H 2 and x = (ω, λ) ∈X we set Γ 2 (g, x) = (ϕ(g, ω, λ), Υ(g, λ)) where Υ(g, λ) is given by (5) and ϕ : H 2 × X 1 × X 2 → X 1 is a map defined by for each g = N k=1 n k s k ∈ H 2 , ω ∈ X 1 and λ = (λ (j) i ) i∈Z,1≤j≤N ∈ X 2 . It is easy to see that the map Γ 2 indeed yields an action of H 2 onX, since the relation ϕ(g + h, ω, λ) = ϕ(h, ϕ(g, ω, λ), Υ(g, λ)) holds for all g, h ∈ H 2 . It is also easy to see that H 1 and H 2 act onX commutatively by Γ 1 and Γ 2 .
By definition the set X (α) (⊂X) is invariant under the action of H 1 . It is also invariant under H 2 . Thus by restricting their domains fromX to X (α) , Γ 1 and Γ 2 yield their actions on X (α) . In what follows we denote them by the same symbols Γ 1 and Γ 2 for simplicity.
acts as a sub-lattice of X 1 by translation.
(iii) The action on the X 1 -part is independent of the X is a subgroup of G 1 , hence follow items (i) and (ii). Consider item (iii). The X 1 -part is given by the map ϕ in (8). We show that for each g = and ω ∈ X 1 the image ϕ(g, ω, λ) of the map ϕ is common to all λ ∈ X (α) 2 . In fact for any λ ∈ X 2 we have Note that if λ lies in X (4) and (6). Hence ϕ(g, ω, λ) is independent of the choice of λ ∈ X (α) 2 .
We define F (α) = (F (α) j,k ) 1≤j,k≤N to be an N × N matrix whose elements are given by and regard ω, n = t (n 1 , . . . , n N ) and ϕ(g, ω, λ) as column vectors. Then (9) is written as ϕ(g, ω, λ) = ω + F (α) n when λ lies in X as a subgroup of S(X 1 ) rather than that of S(X (α) ). In this sense the group G (α) 2 is isomorphic to the lattice F (α) Z N which acts on X 1 = Z N by translation.
Thus Theorem 1 for the present construction with Lemma 7 (to appear in the appendix) is now stated as follows.
Proposition 4. Let the set X (α) , the matrix F (α) , the groups H 1 , H 2 and their actions on X (α) be def ined as above, and x be any element of X (α) . Then there is a bijection between H 1 · [x] H 2 and F (α) Z N \Z N that is commutative with the action of H 1 .
The set F (α) Z N \Z N becomes compact (and is a torus) if and only if det F (α) = 0. A way to achieve this condition is given as follows. Let L be an integer satisfying L ≥ 2 N k=1 km k and set B j,k = 2 min(j, k).
Note that p N = 0 is allowed here. In this case we have Λ(m N , p N ) = Λ (m N ) (m N , p N ) which has a unique element (0) i∈Z . Example 5. The p j and m j in Example 1 satisfy the relation (11) with L = 24. We set α = (α 1 , α 2 , α 3 ) = (3, 1, 1) as in Example 3. By taking B j,k as in (12)   . . , j s }. A generalization of the arguments in this section is given by replacing the positive integer sequence (m j ) 1≤j≤N by (m j ) j∈H , which fits to the case in [5].
All the results in this paper can be extended to this generalized setting: We set X 1 = Z s , and H 2 to be a free abelian group with generators {s j } j∈H . Given any positive integer sequence (p j ) j∈H we set X 2 = j∈H Λ(m j , p j ). Proposition 4 becomes a claim on the bijection between j,k ) j,k∈H is now an s × s matrix. The other definitions and statements can be modified similarly.

A one-dimensional integrable cellular automaton 4.1 The periodic box-ball system
The periodic box-ball system (pBBS) is a one-dimensional cellular automaton with periodic boundary conditions. We give a brief review on this system based on [5] from here to Subsection 4.3. Let L be a positive integer and p be a sequence of letters 1 and 2 under the conditions #(1) ≥ #(2) and #(1) + #(2) = L. Such sequences are called paths of positive weight and of length L. Denote by P the set of all such paths. We can define a commuting family of time evolutions T k (k = 1, 2, . . .) acting on P. It is a collection of update procedures for the cellular automaton. In this paper we write nT k for an n times repeated application of T k instead of T n k , regarding T k s as generators of an abelian group (see Subsection 4.3).
The action of T 1 is given by a cyclic shift by one digit to the right. The definition of T k for the other k's is given by means of the crystal basis of the quantized envelope algebra [11] and is available in Section 2.2 of [5]. Here we review it shortly.

Proposition 5 ([5])
. Given any b 1 × · · · × b L ∈ P ⊂ B ×L 1 , let v 0 ∈ B k be the one defined in the same way as (13) by Then we have v = v ′ in (13) when we adopt this v 0 as the v there.
By this choice of v, we define the time evolution T k by Example 7. We have T 3 (1122211112212) = 2211122211121 by Example 6.
Remark 4. Given any initial condition, the actions of T k s for sufficiently large k's are common to them and determine a unique time evolution. The original study of pBBS [2,3] considers only this time evolution which we denote by T ∞ . In the context of the dynamical system in Subsection 2.3, we can regard the T ∞ as the unique time evolution in this dynamical system, and the other T k s can be thought of as the phase flows associated with the other first integrals.
When sufficiently separated from the other 2's, one can think of a consecutive sequence of 2's of length k as a soliton of amplitude k. By an appropriate definition one can say that the number of solitons conserves for each amplitude. In this example there are four solitons of distinct amplitudes (1, 2, 3 and 4) in every time step. For instance, in the state for t = 9 in the above example, the sequence 2 2 2 1 1 2 2 2 should be interpreted as an intermediate stage of the collision of two solitons of amplitudes 4 and 2.

Soliton content and rigged configurations
Let P + be the set of all ballot sequences in P: A path p = b 1 b 2 . . . b L is called a ballot sequence if and only if the number of 1's in the prefix b 1 . . . b k is at least as large as the number of 2's in that prefix for every k. For any p ∈ P there exist an integer d and a sequence p + ∈ P + such that p = (dT 1 ) · p + where T 1 is the right cyclic shift operator.
Given any p + ∈ P + , a bijective map φ due to Kerov-Kirillov-Reshetikhin [9,10]  Suppose there are two possible ways to write p as p = (dT 1 ) · p + = (d ′ T 1 ) · p ′ + . Then it can be shown that one obtains a common configuration m by φ(p + ) = (m, J ) and φ(p ′ + ) = (m, J ′ ) with some J , J ′ . Hence for each p ∈ P one can label it with a specific m, and we say that such a path has soliton content m. The set P is decomposed as P = m P(m) where P(m) is the set of all paths of soliton content m. It can be shown that each P(m) is invariant under the action of T k for all k.

Direct scattering transform
By means of the map φ one can construct a bijection between the set of all states in the cellular automaton and the set of extended rigged configurations divided by a group action. The map from the former to the latter is called the direct scattering transform (and its inverse is referred to as the inverse scattering transform), named after a similar transform in the theory of integrable non-linear evolution equations [12]. For the sake of simplicity we assume H = {1, . . . , N } in what follows. Then we have m = (m j ) 1≤j≤N , J = ((J (j) i ) 1≤i≤m j ) 1≤j≤N and the vacancy numbers are given by (11). See Remark 3 for the recipe to recover the original setting for general H.
We define and J (m) = N j=1 J (m j , p j ). There is a map where the infinite sequence (J In this article we call the elements of J (m) extended rigged configurations. The relation between J (m) and the setX in Subsection 3.1 will be explained in the next subsection.
Let A be a free abelian group with generators σ 1 , . . . , σ N . We define its action on the set J (m). Given any g = N k=1 n k σ k ∈ A and J = (J .
Recall the family of time evolutions T k (k = 1, 2, . . .) introduced in Subsection 4.1. Let T be the free abelian group generated by T 1 , . . . , T N which acts on P(m). We define its actions on the set J (m). Given any h = N k=1 n k T k ∈ T and J = (J . The direct scattering transform Φ ((3.15) in [5]) is given as follows: Here the second map is given by φ together with the map ι (15), where for any d ∈ Z and J = ((J . As in Subsection 4.2, suppose there are two possible ways to write p as p = (dT 1 ) · p + = (d ′ T 1 ) · p ′ + . Then one generally has ι(J ′ )+ d ′ = ι(J )+ d in J (m). However it can be shown that the relation [ι(J ′ ) + d ′ ] A = [ι(J ) + d] A holds, permitting the map Φ to be well-defined.
Example 9. Let J and J ′ be the ones in Example 8. Then we have (σ 1 +σ 2 )·(ι(J )+d) = ι(J ′ )+ d ′ with d = 0, d ′ = 6. Hence for the p and m in that example, Φ(p) = [ι(J ) It is easy to see that the actions of A and T on J (m) is commutative. Hence by Lemma 1 there is a natural action of T on A\J (m).
The main result of [5] is as follows.
Proposition 6 ([5, Theorem 3.12]). The map Φ is a bijection between P(m) and A\J (m) that is commutative with the action of T .

Construction of the invariant tori
Let α be a common divisor of m and p. As a subset of J (m, p) we define Choose a sequence α = (α j ) 1≤j≤N with each α j being a common divisor of m j and p j . We define . This α represents the symmetry of the system that depends on the initial conditions. As a subset of J (m) it is invariant under the actions of A and T . Let . As a corollary of Proposition 6 we have the following.

Corollary 1.
The map Φ is a bijection between P (α) (m) and A\J (α) (m) that is commutative with the action of T .
be the set introduced in (7). In what follows we identify the group H 2 in Subsection 3.1 with the group A by s i = σ i , and set p j and B j,k as in (11) and (12).

Lemma 5. There is a bijection between J
(α) (m) and X (α) that is commutative with the action of A.
By this lemma we can identify the setX = X 1 × X 2 in Subsection 3.1 with the set J (m).
By Lemmas 5 and 8 (to appear in the appendix) there is a natural action of T on X (α) that is commutative with the action of A. Also one has a bijection between A\J (α) (m) and A\X (α) that is commutative with the action of T . Now by Corollary 1 we have the following.

Proposition 7.
There is a bijection between P (α) (m) and A\X (α) that is commutative with the action of T .
We denote by Ψ the map representing this bijection. Then for any p ∈ P (α) (m) there exists some x ∈ X (α) such that Ψ(p) = [x] A . Now as a corollary of Proposition 7 we have the following. Proposition 8. Given p ∈ P (α) (m) and x ∈ X (α) as above, there is a bijection between T · p(⊂ P (α) (m)) and T · [x] A (⊂ A\X (α) ) that is commutative with the action of T .
The action of the abelian group T on the set X (α) = X 1 × X (α) 2 keeps X (α) 2 untouched. Thus this action is essentially defined on the set X 1 = Z N . By Lemma 8 it is uniquely determined from the action of T on the set P(m) in Subsection 4.3. An explicit description of this action is as follows. For each h = N k=1 n k T k ∈ T and ω = (ω (j) ) 1≤j≤N ∈ X 1 we set h · ω = ω + N k=1 n k h k where h k = (min(j, k)) 1≤j≤N ∈ X 1 . Lemma 6. The action of the group T on the set X 1 = Z N is transitive.
Remark 5. For the generalized case in Remark 3 we define T to be the free abelian group generated by T 1 , T j 1 +1 , . . . , T j s−1 +1 . Then one can show that the action of T on the set X 1 = Z s is transitive in the same way as above.
By Lemma 6 one can adopt T as the abelian group H 1 in Subsection 3.1. Given any p ∈ P (α) (m), a path of soliton content m and with the symmetry α, one can obtain the matrix F (α) by (10), (11) and (12). By Propositions 4 and 8 we finally obtain our second main result.

Theorem 2.
There is a bijection between T · p and F (α) Z N \Z N that is commutative with the action of T .
As we have shown at the end of Subsection 3.3 the F (α) Z N \Z N is an N -dimensional discrete torus. By the considerations given just above Lemma 6 one also knows that each time evolution T k yields a straight motion on the torus with the "velocity" vector h k = (min(j, k)) 1≤j≤N . there is a bijection between T · p and F (α) Z 3 \Z 3 . This is a consequence of Examples 5, 8 and 10.
5 Periods and the level set structure

Dynamical periods
Given p ∈ P (α) (m), a path of soliton content m and with the symmetry α, there is a smallest positive integer n k such that (n k T k ) · p = p for each k. It is called the dynamical period of p associated with the time evolution T k . Since the time evolutions are mapped to straight motions on the torus F (α) Z N \Z N , it is easy to obtain an explicit formula for n k .
We fix a bijection Ψ from T · p to F (α) Z N \Z N whose existence was ensured by Theorem 2. Then since p lies in T · p there exists ω ∈ Z N such that Ψ(p) = [ω] F (α) Z N . By the definition of the action of T on F (α) Z N \Z N we have On the other hand since Ψ is commutative with the action of T we have Thus n k h k must lie in the F (α) Z N -orbit of 0, implying that n k is defined as the smallest positive integer satisfying n k h k ∈ F (α) Z N . This condition is essentially the same one given in (4.32) of [5] which yields an explicit formula for this quantity ((4.26) of [5]). Note that our formulation uses no Bethe ansatz considerations to obtain this result.
Example 12. For p = 121122111212211222121111 one can observe the dynamical periods n 1 = 24, n 2 = 48 and n 3 = 72 by directly applying the time evolutions in Subsection 4.1. By using the above considerations they are also calculated as , det F (α) det F (α) [3] ,

Decomposition of the level set
Each element λ = (λ i ) i∈Z of the set Λ(m, p) is specified by a non-decreasing sequence of integers (0 ≤)λ 2 ≤ λ 3 ≤ · · · ≤ λ m (≤ p), or a partition (λ m , . . . , λ 2 ). Hence the cardinality of Λ(m, p) is given by the binomial coefficient |Λ(m, p)| = p+m−1 m−1 . Recall the decomposition of the set Λ(m, p) = α Λ (α) (m, p), where α runs over every common divisor of m and p. Thus we have the relation The formula (1) for the cardinality of the whole level set with soliton content m is now written as This expression for the volume formula shows the decomposition of the level set (2) into the invariant tori: For each α = (α j ) 1≤j≤N we have the tori F (α) Z N \Z N with the multiplicity 1≤j≤N |Λ (α j ) (m j ,p j )| m j /α j . See [13, Lemmas 3.1 and 3.2] for a proof of this statement. While the smallest torus is given by α = (α j ) 1≤j≤N with every α j being the greatest common divisor of m j and p j , the largest torus is given by α = (1) 1≤j≤N when p N > 0, or α = (1, . . . , 1, m N ) when p N = 0.
One can check that this expression for the multiplicity is indeed an integer. Let s be the generator of the cyclic group C m that acts on Λ (α) (m, p). For each λ ∈ Λ (α) (m, p) the m/α elements (ks) · λ (0 ≤ k ≤ m/α − 1) are all distinct and hence |Λ (α) (m, p)| is divisible by m/α.
An expression for the quantity |Λ (α) (m, p)| is given as follows. Let µ(n) be the Möbius function of number theory [14]; that is, µ(1) = 1, µ(n) = 0 if n is divisible by the square of an integer greater than one, and µ(n) = (−1) r if n is the product of r distinct primes. Then we have where β runs over every common divisor of m and p that is a multiple of α.

A Two elementary lemmas
The following lemma is used in Subsection 3.3 Lemma 7. Let G, H be groups and X be a set. Suppose there are commutative actions of G and H on X. Denote by Γ (resp. Γ ′ ) the action of G (resp. H) on X, and by ρ Γ (resp. ρ Γ ′ ) its permutation representation. Then: (i) The actions of ρ Γ (G) and H on X are commutative.
(ii) There is a bijection between G\X and ρ Γ (G)\X that is commutative with the action of H.
(iii) For all x ∈ X there is a bijection between H ·[x] G and ρ Γ ′ (H)·[x] ρ Γ (G) that is commutative with the action of H.
Proof . (i) For all g ∈ G, h ∈ H and x ∈ X we have h · (ρ Γ (g) · x) = h · Γ (g, x) = Γ (g, h · x) = ρ Γ (g) · (h · x). (ii) Given any p ∈ G\X it can be written as p = [x] G with some x ∈ X. Define ψ : G\X → ρ Γ (G)\X by the relation ψ(p) = [x] ρ Γ (G) . It is easy to see that this map is well-defined and yields the desired bijection. (iii) The bijection is given from that in (ii) by restricting its domain to H · [x] G .
The following lemma is used in Subsection 4.4 Lemma 8. Let G be a group and X 1 , X 2 be sets. Suppose there are actions of G on X 1 and X 2 , and there is a bijection φ : X 1 → X 2 that is commutative with the actions of G. Let H be another group, and suppose there is an action of H on X 1 that is commutative with the action of G. Then: (i) There is a unique action of H on X 2 that is commutative with φ, and commutative with the action of G on X 2 .
(ii) There is a bijection between G\X 1 and G\X 2 that is commutative with the action of H.
Proof . (i) For all h ∈ H and x ∈ X 2 the desired action is uniquely determined by h · x = φ(h · φ −1 (x)). (ii) Given any p ∈ G\X 1 it can be written as p = [x] G with some x ∈ X 1 . Definē φ : G\X 1 → G\X 2 by the relationφ(p) = [φ(x)] G . It is easy to see that this map is well-defined and yields the desired bijection.

B An algorithm for Kerov-Kirillov-Reshetikhin map
We present an algorithm for the map of Kerov-Kirillov-Reshetikhin based on [15]. Let p = b 1 . . . b L be a ballot sequence of letters 1 and 2. We define M 1 , M 2 to be a pair of subsets of {1, . . . , L} that is specified by the following condition: The integer j lies in M 1 (resp. M 2 ) if and only if b j = 1 and b j+1 = 2 (resp. b j = 2 and b j+1 = 1) . Here we interpret b L+1 = 1.
We define Op 1 , Op 2 to be a pair of operators acting on finite sets of distinct integers: Given any such set M , the Op 1 (resp. Op 2 ) replaces its i-th smallest element, say x, by x − (2i − 1) (resp. x − 2i) from i = 1 to i = |M |.
Let ϕ = {} be an empty data set. We repeat the following procedure until we have M 1 = M 2 = ∅.