Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice
CCombinatorial aspects of the conserved quantities ofthe tropical periodic Toda lattice
Taichiro Takagi
Department of Applied Physics, National Defense Academy, Kanagawa 239-8686,Japan
Abstract.
The tropical periodic Toda lattice (trop p-Toda) is a dynamical systemattracting attentions in the area of the interplay of integrable systems and tropicalgeometry. We show that the Young diagrams associated with trop p-Toda given bytwo very different definitions are identical. The first definition is given by a Laxrepresentation of the discrete periodic Toda lattice, and the second one is associatedwith a generalization of the Kerov-Kirillov-Reshetikhin bijection in the combinatoricsof the Bethe ansatz. By means of this identification, it is shown for the first time thatthe Young diagrams given by the latter definition are preserved under time evolution.This result is regarded as an important first step in clarifying the iso-level set structureof this dynamical system in general cases, i. e. not restricted to generic cases.
1. Introduction
The Toda lattice is one of the most famous integrable systems in classical mechanics[1]. Recently, one of its variations is attracting attentions in the context of connectionsbetween tropical geometry and integrable systems [2]. We call this system the tropicalperiodic Toda lattice (trop p-Toda) [3, 4, 5]. Its evolution equation was known asthe ultra-discretization of the discrete periodic Toda equation [6]. In [7, 8], Inoue andTakenawa studied this system and clarified its iso-level set structure under a certaincondition which they call generic . From the viewpoint of tropical geometry, thiscondition is related to the smoothness of the tropical spectral curve determined bythe conserved quantities of the system.In this paper we study conserved quantities of trop p-Toda without the genericcondition. In particular, we show that the Young diagrams associated with trop p-Todagiven by two very different definitions are identical. From one of the definitions oneimmediately sees that the common Young diagram is preserved under time evolution.In the context of integrable cellular automaton explained below, this Young diagramrepresents the content of solitons in the system, and the generic condition is requiringno two solitons have a common amplitude. We believe that this identification of theYoung diagrams is the first step to clarify the iso-level set structure of this dynamicalsystem in general cases, i. e. not restricted to generic cases. a r X i v : . [ n li n . S I] M a r ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice k th conservedquantity of dp-Toda is defined as a sum of products of k dependent variables whoseindices obey a nearest-neighbor-exclusion condition. Then the corresponding conservedquantity of trop p-Toda is defined as its tropical limit or tropicalization [9, 10, 11].We show that the above condition leads to a condition of weak convexity relatingthe conserved quantities, which is also a new result of this paper that enables us torepresent them as a Young diagram. The second is related to a generalization of theKerov-Kirillov-Reshetikhin (KKR) bijection in combinatorics of Bethe ansats [12, 13],especially one of its variations in sl case [14]. It is also considered as a continuousanalogue of the ‘10-elimination’ algorithm for conserved quantities of an integrablecellular automaton known as the periodic box-ball system (pBBS) [15].We note that for a special case there has already been attention paid to thisremarkable equivalence of Young diagrams. The pBBS is regarded as a case of tropp-Toda, where the values of its dependent variables are restricted to positive integers.In this case, the equivalence of the Young diagrams has been pointed out by Iwao andTokihiro [16]. On the basis of their idea of drawing diagrams associated with the seconddefinition of the Young diagram, we give a proof of our main theorem for the conservedquantities of general trop p-Toda.Here we explain the reason why we expect that the identification of the Youngdiagrams from Lax representation and those from generalized KKR bijection is the firststep to clarify the iso-level set structure of trop p-Toda without the generic condition.In the pBBS case, the KKR bijection gives the action-angle variables of this dynamicalsystem [17]. While the action variables are the conserved quantities, the angle variablesyield certain time evolutions which turn out to be flows on the iso-level set. By thisfact the author has succeeded to clarify the iso-level set structure of pBBS withoutthe generic condition [18]. Since the trop p-Toda is a generalization of the pBBS, it isreasonable to consider the corresponding generalization of the KKR bijection. We notethat in the pBBS case the conservation of the Young diagrams defined by KKR bijectionis directly proved by using the crystal theory, combinatorial R maps, and Yang-Baxterrelations (See Theorem 2.2 and Proposition 3.4 of [17]). Since these methods are notdeveloped in trop p-Toda case, our main result of this paper is so far the only proof ofthe conservation of the Young diagrams defined by the generalized KKR.Readers may wonder why trop p-Toda is worth studying independently of thealready known many results in pBBS. The most remarkable difference between pBBSand trop p-Toda is that the iso-level set of the latter is not a finite set but is analgebraic variety. This implies that while any state comes back to the same state inpBBS, that is not true in trop p-Toda. Actually, when the lengths of the solitons arelinearly independent over the field of rational numbers, the phase flow can be dense inthe iso-level set as in the case of classical mechanics [19]. However, this is nothing butonly one aspect of the fact that their iso-level sets are totally different mathematicalobjects. A really important problem here is that the structure of the iso-level set of trop ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice § § § § §
3. In § § § § § § §
4. Some concluding remarks are given in §
2. Discrete and tropical periodic Toda lattice
Throughout this paper we use the symbol (cid:47) by the following meaning: i (cid:47) j ⇔ i + 1 < j. (1)Given a sequence of real numbers a (= 0) , a , a , . . . , let c ( N ) k be the numbers defined bythe recursion relation c ( N ) k = c ( N − k + ( a N − + a N ) c ( N − k − − a N − a N − c ( N − k − , (2)and the boundary conditions c ( N ) k = 0 for k < k > N, c ( N )0 = 1 for N ≥ . (3)Then it is easy to see that the unique solution of (2) under (3) is given by c ( N ) k = (cid:88) ≤ i (cid:47) i (cid:47) ··· (cid:47) i k ≤ N a i a i · · · a i k . (4) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice e ( N ) k be the numbers defined by the relations e ( N )1 = c ( N )1 , e ( N )2 = c ( N )2 − a a N , (5) e ( N ) k = c ( N ) k − a a N (cid:88) ≤ i (cid:47) i (cid:47) ··· (cid:47) i k − ≤ N − a i a i . . . a i k − , (6)for 3 ≤ k ≤ N . Then it is easy to see that e ( N ) k = (cid:88) ≤ i (cid:47) i (cid:47) ··· (cid:47) ik ≤ N ( i ,i k ) (cid:54) =(1 , N ) a i a i . . . a i k . (7)Let F ( x ; a , a ) = x + a + a and F N ( x ; a , . . . , a N ) = det x + a + a a a x + a + a a a . . . . . .. . . . . . 1 a N − a N − x + a N − + a N , (8)for N ≥ Lemma 1 ([20], Proposition 7.1) F N ( x ; a , . . . , a N ) = N (cid:88) k =0 c ( N ) k x N − k . (9) Proof.
Let F N ( x ; a , . . . , a N ) = (cid:80) Nk =0 b ( N ) k x N − k . Then we have b ( N )0 = 1. By expanding(8) with respect to its N th row one obtains F N ( x ; a , . . . , a N ) = ( x + a N − + a N ) F N − ( x ; a , . . . , a N − ) − a N − a N − F N − ( x ; a , . . . , a N − ) . (10)Defining b ( N ) k = 0 for k < k > N , one can deduce from (10) that the b ( N ) k ’s satisfythe same recursion relation (2) as c ( N ) k ’s for 1 ≤ k ≤ N . Hence b ( N ) k = c ( N ) k . (cid:3) Let G N ( x ; a , . . . , a N ) = det x + a + a − N − a a N /ya a x + a + a a a . . . . . .. . . . . . 1( − N − y a N − a N − x + a N − + a N , (11)for N ≥ Lemma 2 G N ( x ; a , . . . , a N ) = y + ( N (cid:89) i =1 a i ) /y + N (cid:88) k =0 e ( N ) k x N − k . (12) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice Proof.
By expanding (11) with respect to its N th column one obtains G N ( x ; a , . . . , a N ) = ( x + a N − + a N ) F N − ( x ; a , . . . , a N − )+ y − a N − a N − F N − ( x ; a , . . . , a N − )+ ( a . . . a N ) /y − a a N F N − ( x ; a , . . . , a N − )= F N ( x ; a , . . . , a N ) − a a N F N − ( x ; a , . . . , a N − )+ y + ( a . . . a N ) /y, (13)where we have used (10). Let G N ( x ; a , . . . , a N ) = y + ( (cid:81) Ni =1 a i ) /y + (cid:80) Nk =0 g ( N ) k x N − K .From Lemma 1 and (13), one finds that the g ( N ) k ’s satisfy the same relations, (5) and(6), as the e ( N ) k ’s for 1 ≤ k ≤ N . Hence g ( N ) k = e ( N ) k . (cid:3) On the basis of [3], we briefly review the derivation of discrete and tropical periodicToda lattice equations. Let { x n ( t ) } n ∈ Z N be a set of smooth functions of time t ∈ R .Set a n = a n ( t ) := 1 + δ ˙ x n ( t ) , a n +1 = a n +1 ( t ) := δ e x n +1 ( t ) − x n ( t ) with δ > a j = a j ( t + δ ) for j ∈ Z N . Then we have lim δ → δ (¯ a n ¯ a n +1 − a n +1 a n +2 ) = 0.Suppose x n = x n ( t )’s satisfy the Toda lattice equation¨ x n = e x n +1 − x n − e x n − x n − . (14)Then we have lim δ → δ (¯ a n − + ¯ a n − a n − a n +1 ) = 0. Under this consideration, wedefine the evolution equations for the discrete periodic Toda lattice as¯ a n − + ¯ a n = a n + a n +1 , ¯ a n ¯ a n +1 = a n +1 a n +2 , (15)where the a n = a tn , ¯ a n = a t +1 n are dependent variables which depend on discrete spatialcoordinate n ∈ Z N and discrete time t ∈ Z . Obviously, (cid:81) Nl =1 a l is a conserved quantity.By lemma 4 we will find that h N = (cid:81) Nl =1 a l − + (cid:81) Nl =1 a l is also a conserved quantity.This implies that ( (cid:81) Nl =1 ¯ a l − , (cid:81) Nl =1 ¯ a l ) = ( (cid:81) Nl =1 a l , (cid:81) Nl =1 a l − ) or ( (cid:81) Nl =1 a l − , (cid:81) Nl =1 a l ).While the former leads to the trivial solution ¯ a n = a n +1 , the latter to a non-trivialsolution ¯ a n = a n +1 + a n − (cid:81) Nl =1 ( a l − /a l ) (cid:80) N − k =0 (cid:81) kl =1 ( a n − l )+1 /a n − l ) ) , ¯ a n +1 = a n +1 a n +2 / ¯ a n . (16)For a derivation of this solution, see Proposition 6.13 of [3]. Now we consider itstropicalization, which is a procedure to replace × by +, and + by min. Note thatthe numerator in (16) is a conserved quantity. By regarding it as a positive constantand setting it to be zero under the tropicalization with trivial valuation [11], we obtaina dynamical system given by the piecewise linear evolution equations¯ A n = min (cid:32) A n +1 , A n − min ≤ k ≤ N − (cid:32) k (cid:88) l =1 ( A n − l )+1 − A n − l ) ) (cid:33)(cid:33) , ¯ A n +1 = A n +1 + A n +2 − ¯ A n , (17) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice T = (cid:40) ( A n ) n ∈ Z N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) l =1 A l < N (cid:88) l =1 A l − (cid:41) ⊂ R N . We call this system the tropical periodic Toda lattice [3].
Remark 3
We have changed the notations as a n − = q n , a n − = w n , A n − = Q n , A n − = W n from those in [3], since this enables us to describe the conservedquantities neatly. Without loss of generality, we can assume all the A -variables in (17) take their valuesin R > . This enables us to represent the time evolution of trop p-Toda by a sequenceof two-colored (white and black) strips, where the lengths of the white (resp. black)segments are denoted by A n − s (resp. A n s). See Figure 1 for an example. t A t A t A t A t A t A Figure 1.
A representation for the time evolution of trop p-Toda. Let t be the time forthe top row, where the lengths of three black segments (‘solitons’) are A t , A t , A t , andthe lengths of white segments between them (with regard to the periodic boundary)are A t , A t , A t . In the second row the length of the first black segment to the right ofthe j -th ‘soliton’ at time t is A t +12 j . In the same way, the strip at the n -th row visualizesthe quantities A t + n − j . Now on the basis of [7] we briefly review the Lax representation for the discreteperiodic Toda lattice equation. Let R ( λ ) = a /λa a . . .. . . 1 a N − , M ( λ ) = a a
1. . . . . . a N − λ a N . (18)We denote by R ( λ ) , M ( λ ) the matrices obtained from R ( λ ) , M ( λ ) by replacing a i by¯ a i for all i . Then the evolution equations of the discrete periodic Toda lattice (15)are equivalent to R ( λ ) M ( λ ) = M ( λ ) R ( λ ). Let L ( λ ) = R ( λ ) M ( λ ) , L ( λ ) = R ( λ ) M ( λ ). ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice L ( λ ) = M ( λ ) L ( λ ) M ( λ ) − . (19)This implies that the polynomial det( x I + L ( λ )) is invariant under the time evolution.Hence its coefficients are conserved quantities. Since G N ( x ; a , . . . , a N ) in (11) isexpressed as G N ( x ; a , . . . , a N ) = det( x I + L (( − N − y )) we have the following: Lemma 4
The discrete periodic Toda lattice (15) has N + 1 conserved quantities h k = (cid:88) ≤ i (cid:47) i (cid:47) ··· (cid:47) ik ≤ N ( i ,i k ) (cid:54) =(1 , N ) a i a i . . . a i k , for ≤ k ≤ N and h N +1 = (cid:81) Ni =1 a i . By means of their tropicalization we obtain:
Lemma 5
The tropical periodic Toda lattice (17) has N + 1 conserved quantities H k = min ≤ i (cid:47) i (cid:47) ··· (cid:47) ik ≤ N ( i ,i k ) (cid:54) =(1 , N ) ( A i + A i + · · · + A i k ) , (20) for ≤ k ≤ N and H N +1 = (cid:80) Ni =1 A i .Proof. As was explained in § a i = e − A i /ε with ε >
0, apply the map x (cid:55)→ − ε log x and take the limit ε →
0. This procedure transforms (16) to (17), whichimplies the claim of this lemma of the basis of the previous one. (cid:3)
Remark 6
We derived Lemmas 4 and 5 directly through the Lax representation. Anequivalent result was obtained by using a different method in [16], Proposition 3.9.2.3. The weak convexity condition relating the conserved quantities
The iso-level set structure of trop p-Toda has been clarified by means of tropicalgeometry [7, 8] in case when the strong convexity condition H k + H k +2 > H k +1 (whichthey call generic) is satisfied by the conserved quantities. In this section we prove thatin general cases only the weak convexity condition H k + H k +2 ≥ H k +1 holds. For thispurpose we first consider a lemma in elementary combinatorics.Put two kinds of symbols ◦ ’s and • ’s on a circle. Say two ◦ ’s are adjacent if thereare no other ◦ ’s between them. Lemma 7
Put k ◦ ’s and k + 2 • ’s on a circle so that their positions do not coincideand there are at most two • ’s between adjacent ◦ ’s. Then on the circle we always havesuch a configuration − • − • − ( ◦ − • ) n − •− for some n ≥ .ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice Proof.
If there exist more than one ◦ ’s between any adjacent • ’s on the circle, removethe ◦ ’s until there remains only one. Suppose the number of ◦ ’s we have removed is α . Since the number of ◦ is now k − α , the number of the configuration − • − • − onthe circle is ( k + 2) − ( k − α ) = α + 2. By construction, we have such configurations − • − • − ( ◦ − • ) n − •− for some n ≥ α + 2) adjacent − • − • − ’s, and atleast two of them will be left unchanged when we put all ( α ) removed ◦ ’s back into theoriginal positions. (cid:3) Example 8
See Figure 2.
Figure 2.
Examples for Lemma 7
Now we present one of our new results in this paper.
Theorem 9
For x = ( x , . . . , x N ) ∈ ( R > ) N let H ( x ) = 0 and H k ( x ) = min ≤ i (cid:47) i (cid:47) ··· (cid:47) ik ≤ N ( i ,i k ) (cid:54) =(1 , N ) ( x i + x i + · · · + x i k ) , (21) for ≤ k ≤ N . Then the relations H k ( x ) + H k +2 ( x ) ≥ H k +1 ( x ) are satisfied for ≤ k ≤ N − .Proof. Suppose we have H k ( x ) = x i + · · · + x i k and H k +2 ( x ) = x j + · · · + x j k +2 . Let S = { i , . . . , i k } and S = { j , . . . , j k +2 } be the sets of their indices. To begin with weassume S ∩ S = ∅ . For a, b ∈ { , . . . , N } say b is next to a if | a − b | = 1 or 2 N − b ∈ S such that b is not next to a for any a ∈ S , then we have H k ( x ) + x b ≥ H k +1 ( x ) and H k +2 ( x ) − x b ≥ H k +1 ( x ). Hence the claim follows.Suppose otherwise, i. e. we assume every b ∈ S is next to some a ∈ S . Draw a circleof circumference 2 N with a spacial coordinate 1 , . . . , N assigned counter-clockwise onit. Put ◦ ’s on the circle at the positions given by S , and put • ’s at those given by S .Since every b ∈ S is next to some a ∈ S , there exist at most two • ’s between adjacent ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice ◦ ’s on the circle. Hence by Lemma 7 we have such a configuration − • − • − ( ◦ − • ) n − •− for some n ≥ − • − ◦ − ( • − ◦ ) n − •− . Nowlet I (resp. I ) be the set of the positions of ◦ ’s (resp. • ’s) on the circle. Then since | I | = | I | = k + 1 we have H k ( x ) + H k +2 ( x ) = (cid:80) i ∈ I x i + (cid:80) i ∈ I x i ≥ H k +1 ( x ).When S ∩ S (cid:54) = ∅ but not S ⊂ S , we can prove the statement by replacing S i by S i \ ( S ∩ S ) for i = 1 , S ∩ S are next to any a ∈ S i \ ( S ∩ S ) for i = 1 ,
2. The case S ⊂ S issimpler, because no elements of S \ S are next to any a ∈ S . The proof is completed. (cid:3) Recall the conserved quantity of the tropical periodic Toda lattice (20). Let˜ l i = H N +1 − i − H N − i for 1 ≤ i ≤ N . Then by Theorem 9 we have ˜ l ≥ · · · ≥ ˜ l N >
0. Let { l i } ≤ i ≤ s be the set of real numbers satisfying l > · · · > l s such that for any 1 ≤ j ≤ N there exists 1 ≤ i ≤ s such that ˜ l j = l i . And let m i = { j | ≤ j ≤ N, ˜ l j = l i } . Nowwe can express the conserved quantities by a Young diagram (Figure 3) in which thelengths of the horizontal edges are not necessarily integers. Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice N with a spacial coordinate1 , . . . , N assigned counter-clockwise on it. Put ◦ ’s on the circle at the positions givenby S , and put • ’s at those given by S . Since every b ∈ S is next to some a ∈ S ,there exist at most two • ’s between adjacent ◦ ’s on the circle. Hence by Lemma 7we have such a configuration − • − • − ( ◦ − • ) n − •− for some n ≥ − • − ◦ − ( • − ◦ ) n − •− . Now let I (resp. I ) be the setof the positions of ◦ ’s (resp. • ’s) on the circle. Then since | I | = | I | = k + 1 we have H k ( x ) + H k +2 ( x ) = ∑ i ∈ I x i + ∑ i ∈ I x i ≥ H k +1 ( x ).When S ∩ S ̸ = ∅ , we can prove the statement by replacing S i by S i \ ( S ∩ S ) for i = 1 , S ∩ S arenext to any a ∈ S i \ ( S ∩ S ) for i = 1 ,
2. The proof is completed. (cid:3)
Recall the conserved quantity of the tropical periodic Toda lattice (20). Let˜ l i = H N +1 − i − H N − i for 1 ≤ i ≤ N . Then by Theorem 9 we have ˜ l ≥ · · · ≥ ˜ l N >
0. Let { l i } ≤ i ≤ s be the set of real numbers satisfying l > · · · > l s such that for any 1 ≤ j ≤ N there exists 1 ≤ i ≤ s such that ˜ l j = l i . And let m i = { j | ≤ j ≤ N, ˜ l j = l i } . Nowwe can express the conserved quantities by a Young diagram (Figure 3) in which thelengths of the arms are not necessarily integers. l s l s − − l s l − l m s m s − m λ = Figure 3.
The Young diagram.
In the context of integrable cellular automaton, this Young diagram represents thecontent of solitons in the system. From this point of view, we have m i solitons ofamplitude l i for 1 ≤ i ≤ s . We note that the lengths of the black segments in Figure 1are not always identical to the amplitudes of solitons represented by λ , since there areintermediate states of collisions of several solitons. We introduce another algorithm to construct Young diagrams from states of trop p-Toda. This algorithm is related to the KKR bijection in sl case which was appliedto the inverse scattering transform of pBBS [17]. It is also regarded as a continuous Figure 3.
The Young diagram.
In the context of integrable cellular automaton, this Young diagram represents thecontent of solitons in the system. From this point of view, we have m i solitons ofamplitude l i for 1 ≤ i ≤ s . We note that the lengths of the black segments in Figure 1are not always identical to the amplitudes of solitons represented by λ , since there areintermediate states of collisions of several solitons. We introduce another algorithm to construct Young diagrams from states of trop p-Toda. This algorithm is related to the KKR bijection in sl case which was appliedto the inverse scattering transform of pBBS [17]. It is also regarded as a continuousanalogue of the ‘10-elimination’ procedure [15]. We shall prove that the Young diagram ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice x , . . . , x n obeys the highest weightcondition if the inequalities k (cid:88) i =1 ( x i − − x i ) ≥ , (22)are satisfied for 1 ≤ k ≤ n . Fix time t and denote the dependent variables of trop p-Toda A ti by A i . By shifting their indices cyclically, one can make the sequence A , . . . , A N obey the highest weight condition. This is due to the phase space condition under (17);suppose that we have this.Let x (1) = ( x (1)1 , . . . , x (1)2 N (1) ) with x (1) i = A i and N (1) = N . Given an array ofpositive real numbers x ( i ) = ( x ( i )1 , . . . , x ( i )2 N ( i ) ) satisfying the highest weight condition,define y ( i ) = ( y ( i )1 , . . . , y ( i )2 N ( i ) ) by y ( i ) j = x ( i ) j − µ ( i ) where µ ( i ) = min ≤ j ≤ N ( i ) { x ( i ) j } . In thearray of non-negative real numbers y ( i )1 , . . . , y ( i )2 N ( i ) , suppose there are k ( i ) sequences ofzeros. Here we regard a lone zero also as a sequence. We denote by n ( i ) j (1 ≤ j ≤ k ( i ) )the length of the j th sequence of zeros. Let N ( i +1) = N ( i ) − (cid:80) k ( i ) j =1 (cid:100) n ( i ) j / (cid:101) where (cid:100) c (cid:101) isthe smallest integer satisfying (cid:100) c (cid:101) ≥ c . If N ( i +1) = 0 then we stop. Otherwise we definean array x ( i +1) = ( x ( i +1)1 , . . . , x ( i +1)2 N ( i +1) ) of positive real numbers satisfying the highestweight condition by the following procedure.(i) If the first sequence of n ( i )1 zeros in the array y ( i )1 , . . . , y ( i )2 N ( i ) is at the left end, thenremove these zeros. We see that n ( i )1 must be even because the array satisfies thehighest weight condition(ii) For any j such that the j -th sequence of n ( i ) j zeros is between positive neighbors a, b as . . . , a, , . . . , , b, . . . , replace it by . . . , a, b, . . . if n ( i ) j is even, or by . . . , a + b, . . . if n ( i ) j is odd. More precisely, we simply remove n ( i ) j zeros in the former, or in thelatter we further replace the neighbors a, b by a single number a + b .(iii) Suppose the last sequence of n ( i ) k ( i ) zeros is at the right end after a positive neighbor a as . . . , a, , . . . ,
0. Remove these zeros. If n ( i ) k ( i ) is odd then also remove a , and addit to the first element of the array.Let x ( i +1) j be the j th element of the resulting array of 2 N ( i +1) positive integers. By thefollowing lemma we see that the sequence x ( i +1)1 , . . . , x ( i +1)2 N ( i +1) satisfies the highest weightcondition. Lemma 10
The highest weight condition is preserved under the following procedures.(i) Insert or remove two consecutive zeros.(ii) Split any positive term into two positive numbers and insert a zero between them,or remove a zero and join its neighbors into one term.
Suppose N ( i ) > ≤ i ≤ u and N ( u +1) = 0. Let ν ( i ) = N ( i ) − N ( i +1) . Obviously theset of numbers { ( µ ( i ) , ν ( i ) ) } ≤ i ≤ u determines a Young diagram ˜ λ in which the lengths ofthe horizontal edges are positive real numbers (Figure 4). ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice µ (1) µ (2) µ ( u ) ν (1) ν (2) ν ( u ) ˜ λ = Figure 4.
The Young diagram.
Example 11
For N = 6 , let A = 2 √ , A = A = 2 , A = A = A = √ , A = A = A = A = A = A = 1 . Then we have: x (1) = (2 √ , , √ , , , , , √ , , , √ , ,y (1) = (2 √ − , , √ − , , , , , √ − , , , √ − , ,x (2) = (2 √ − , , √ − , √ − , √ − , ,y (2) = ( √ , − √ , , , , − √ ,x (3) = ( √ , − √ ,y (3) = (3 √ − , , and µ (1) = 1 , µ (2) = √ − , µ (3) = 4 − √ , ν (1) = 3 , ν (2) = 2 , ν (3) = 1 . Example 12
For N = 4 , let A = 4 , A = A = 3 , A = A = 2 , A = A = A = 1 .Then we have: x (1) = (4 , , , , , , , ,y (1) = (3 , , , , , , , ,x (2) = (4 , , , ,y (2) = (3 , , , ,x (3) = (4 , ,y (3) = (3 , , and µ (1) = µ (2) = µ (3) = 1 , ν (1) = 2 , ν (2) = ν (3) = 1 . Now we present the main result of this paper.
Theorem 13
For any integer ≤ k ≤ N , the area of the part of the Young diagram ˜ λ between its bottom line and the horizontal line above it at distance k is given by H k , the k -th conserved quantity of the trop p-Toda defined as in (20) . In other words, the Young diagram ˜ λ in Figure 4 coincides with the Young diagram λ in Figure 3. We shall give a proof of this theorem in the next section. Figure 4.
The Young diagram.
Example 11
For N = 6 , let A = 2 √ , A = A = 2 , A = A = A = √ , A = A = A = A = A = A = 1 . Then we have: x (1) = (2 √ , , √ , , , , , √ , , , √ , ,y (1) = (2 √ − , , √ − , , , , , √ − , , , √ − , ,x (2) = (2 √ − , , √ − , √ − , √ − , ,y (2) = ( √ , − √ , , , , − √ ,x (3) = ( √ , − √ ,y (3) = (3 √ − , , and µ (1) = 1 , µ (2) = √ − , µ (3) = 4 − √ , ν (1) = 3 , ν (2) = 2 , ν (3) = 1 . Example 12
For N = 4 , let A = 4 , A = A = 3 , A = A = 2 , A = A = A = 1 .Then we have: x (1) = (4 , , , , , , , ,y (1) = (3 , , , , , , , ,x (2) = (4 , , , ,y (2) = (3 , , , ,x (3) = (4 , ,y (3) = (3 , , and µ (1) = µ (2) = µ (3) = 1 , ν (1) = 2 , ν (2) = ν (3) = 1 . Now we present the main result of this paper.
Theorem 13
For any integer ≤ k ≤ N , the area of the part of the Young diagram ˜ λ between its bottom line and the horizontal line above it at height k is given by H k , the k th conserved quantity of the trop p-Toda defined as in (20) .ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice λ in Figure 4 coincides with the Young diagram λ in Figure 3. We shall give a proof of this theorem in the next section.
3. Proof of the main theorem
By generalizing the method of Appendix A. 1 of [16], we introduce a way to draw agraph Φ A associated with a state of the trop p-Toda A = ( A , . . . , A N ). It has severalconnected components called trees. See Figure 5 for an example, that is for the A -variables in Example 11. At the end we rewrite the assertion of Theorem 13 in termsof the trees.Recall the algorithm in § N = N (1) . Place the symbols A , . . . , A N or x (1)1 , . . . , x (1)2 N (1) at a horizontal level, called level 0. Draw a vertical line from each symbolupwardly to a certain horizontal level, called level 1. We associate the non-negative realnumbers y (1)1 , . . . , y (1)2 N (1) to the lines. If y (1) j = 0, we put a symbol × at the top of theline. Then, change the symbol × into another symbol ⊗ if it is an isolated one, or isat an ‘odd-th’ position of a sequence of consecutive × ’s. In what follows we will payattention to ⊗ ’s only and ignore the other × ’s. Each ⊗ is called the top of a tree oflevel 1. For each isolated ⊗ or sequence of ⊗ ’s placed at every other position, we letthe lines adjacent to ⊗ ’s join together to straddle the ⊗ ’s. Here we respect the periodicboundary condition, so the leftmost and rightmost ends are regarded as adjacent. Wecall each joining point a branching point of level 1. After this procedure, the number oflines is reduced to 2 N (2) .Now we describe a general procedure to draw the diagram from level i − i . We associate the positive real numbers x ( i )1 , . . . , x ( i )2 N ( i ) to the tops of the 2 N ( i ) linesat level i −
1. Extend the lines upwardly to level i . We associate the non-negative realnumbers y ( i )1 , . . . , y ( i )2 N ( i ) to the lines. If y ( i ) j = 0, we put a symbol × at the top of theline. By repeating the same procedure in the previous paragraph, we obtain the tops ofthe trees, as well as the branching points, of level i . A A A A A A A A A A A A Figure 5.
A tree diagram for the A -variables ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice A for the state A = ( A , . . . , A N ). Denote by T ree (Φ A ) the set of all trees in the graph Φ A . Define the level of a tree by the level ofits top point marked by ⊗ . By construction, the number of level i trees is ν ( i ) . Hencethere are (cid:80) ul =1 ν ( l ) = N trees in total. Let t ∈ T ree (Φ A ) be a tree of level i . We defineits height by Ht( t ) = (cid:80) il =1 µ ( l ) . Definition 14
We label all the trees in
T ree (Φ A ) as t , . . . , t N so that their heightsare in weakly increasing order, i. e. i < j ⇒ Ht( t i ) ≤ Ht( t j ) . Let Tree ( k ) = { t , . . . , t k } . Now we see that the assertion of Theorem 13 is equivalent to the relation k (cid:88) i =1 Ht( t i ) = min ≤ i (cid:47) i (cid:47) ··· (cid:47) ik ≤ N ( i ,i k ) (cid:54) =(1 , N ) ( A i + A i + · · · + A i k ) , (23)for 1 ≤ k ≤ N . We present several elementary lemmas on the graph Φ A that are necessary for provingTheorem 13. Given t ∈ T ree (Φ A ), we define a set of indices of the A -variables asRoot( t ) = { i ∈ { , . . . , N }| A i is at a bottom point of t } . (24)For T ⊂ T ree (Φ A ) we write Root( T ) = (cid:70) t ∈ T Root( t ). If P is a branching point of t at level i , we define its height by Ht( P ) = (cid:80) il =1 µ ( l ) . (Note that we regard a tree anda branching point of the same level as having common height, though they are not sodepicted in figures for technical reasons.) Say P has multiplicity m P if there are m P + 2lines outgoing from P . If the tree t has branching points P , . . . , P q with multiplicities m P , . . . , m P q , define the weight of t bywt( t ) = Ht( t ) + q (cid:88) j =1 Ht( P j ) m P j . (25)Then we have: Lemma 15 ([16], Lemma A.1)
For any t ∈ T ree (Φ A ) it holds that wt( t ) = (cid:88) i ∈ Root( t ) A i . (26)The set T ree (Φ A ) becomes a partially ordered set on introducing the following partialorder. Denote by t (cid:31) s when s is straddled by t . We denote by t (cid:23) s a case whereeither t (cid:31) s or t = s is satisfied. For t ∈ T ree (Φ A ) we defineSub( t ) = { s ∈ T ree (Φ A ) | t (cid:31) s } , (27)Sub( t ) = { s ∈ T ree (Φ A ) | t (cid:23) s } = Sub( t ) (cid:116) { t } . (28) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice t ) takes either an ‘odd-th’ position or an ‘even-th’ position in itsnesting structure, regarding t itself as taking the first position and the order of thenesting as increasing inwardly. We denote by Sub o ( t ) ⊂ Sub( t ) the set of all trees atodd-th positions, and let Sub e ( t ) = Sub( t ) \ Sub o ( t ). Accordingly, we defineRoot o ( t ) = Root(Sub o ( t )) , (29)Root e ( t ) = Root(Sub e ( t )) , (30)and Root( t ) = Root o ( t ) (cid:116) Root e ( t ). Then we have: Lemma 16 ([16], Lemma A.2)
For any t ∈ T ree (Φ A ) it holds that (cid:88) s ∈ Sub( t ) Ht( s ) = (cid:88) i ∈ Root o ( t ) A i . (31)We also use the following lemmas in the next subsection. Lemma 17 t ) = o ( t ) . (32) Proof.
Denote all the branching points in Sub( t ) by P , . . . , P r and their multiplicitiesby m P , . . . , m P r . By looking the graph downwardly, we see that the number of treesincreases by m at a branching point with multiplicity m . Hence t ) = 1+ (cid:80) ri =1 m P i .In the same way, we see that the number of vertical lines increases by 2 m at the branchingpoint. Hence t ) = 1 + (cid:80) ri =1 m P i , which implies o ( t ) = 1 + (cid:80) ri =1 m P i . (cid:3) Recall the definition of the set Tree ( k ) in Definition 14. One can prove the followinglemma by induction on k . Lemma 18
For any t ∈ Tree ( k ) the relation Sub( t ) ⊂ Tree ( k ) holds. Say ξ is a maximal element of a partially ordered set X if ξ (cid:23) ξ (cid:48) is satisfied for any ξ (cid:48) ∈ X that is comparable to ξ with respect to the partial order. Definition 19
Let
MTree ( k ) ⊂ Tree ( k ) be the set of all maximal elements of Tree ( k ) . Then we have:
Lemma 20 (cid:71) t ∈ MTree ( k ) Sub( t ) = Tree ( k ) . (33) Proof.
We show the inclusion ⊂ since the opposite inclusion is almost trivial. Suppose u is an element of LHS. Then there exists t ∈ MTree ( k ) ⊂ Tree ( k ) such that u ∈ Sub( t ).This implies u ∈ Tree ( k ) by Lemma 18. (cid:3) Definition 21
Let
MSub( t ) ⊂ Sub( t ) be the set of all maximal elements of Sub( t ) .ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice t ) = (cid:71) s ∈ MSub( t ) Sub( s ) , (34)Root e ( t ) = (cid:71) s ∈ MSub( t ) Root o ( s ) . (35)From (28), (34), (35) and Lemma 16 we have: Lemma 22 (cid:88) i ∈ Root o ( t ) A i = Ht( t ) + (cid:88) i ∈ Root e ( t ) A i . (36)Let t ∈ T ree (Φ A ) be a tree and P ∈ t is one of its branching points. Consider asubtree t ⊂ t that extends downwardly from P . See Figure 6. Define Root( t ) , Sub( t ),etc. by extending the definitions (24), (27), etc. in an obvious way. Note that t itself isnot an element of T ree (Φ A ). Then we have: Lemma 23 (cid:88) i ∈ Root o ( t ) A i (cid:61) Ht( P ) + (cid:88) i ∈ Root e ( t ) A i . (37) Proof.
If the subtree t has branching points Q , . . . , Q s with multiplicities m Q , . . . , m Q s , define its weight bywt( t ) = Ht( P ) + s (cid:88) j =1 Ht( Q j ) m Q j . (38)Then by the algorithm of drawing the graph Φ A we can deduce wt( t ) ≤ (cid:80) i ∈ Root( t ) A i . We define t called the completion of t as a tree obtained by extending the top of t by the length (cid:80) i ∈ Root( t ) A i − wt( t ). In other words t is a tree that shares allthe branching/bottom points with t but satisfies Lemmas 15 and 16. Then the claimfollows by applying Lemma 22 on the tree t and using wt( t ) ≥ wt( P ). (cid:3) Now we give a proof of the main theorem, leaving proofs of two more lemmas to appearafterwards in the following subsections.Given N we define the sets of nearest neighbor excluding indices as B ( k, N ) = {{ i , . . . , i k } ⊂ Z k | ≤ i (cid:47) i (cid:47) · · · (cid:47) i k ≤ N, ( i , i k ) (cid:54) = (1 , N ) } , (39) B ( N ) = (cid:71) (cid:53) k (cid:53) N B ( k, N ) . (40)Then the RHS of (23) can be written as min B ∈B ( k,N ) { (cid:80) i ∈ B A i } . The Forest RealizationLemma in § ≤ k ≤ N there exists T ⊂ T ree (Φ A ) such thatthe relation min B ∈B ( k,N ) (cid:40)(cid:88) i ∈ B A i (cid:41) = (cid:88) i ∈ Root( T ) A i , (41) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice t t P t Figure 6. (left) The subtree t defined as a branch of t under a branching point P :(right) A tree t defined as the completion of t is satisfied. Then, from the Close Packing Lemma in § T must be written by a disjoint union as T = (cid:71) t ∈ U Sub o ( t ) , U ⊂ T ree (Φ A ) . (42)This implies that (cid:88) i ∈ Root( T ) A i = (cid:88) t ∈ U (cid:88) i ∈ Root o ( t ) A i = (cid:88) t ∈ U (cid:88) s ∈ Sub( t ) Ht( s ) ≥ k (cid:88) i =1 Ht( t i ) , (43)where we have used Lemma 16 and the relation (cid:80) t ∈ U t )) = k which was verifiedby Lemma 17. Hence it suffices to show that there exists U ⊂ T ree (Φ A ) such that therelation (cid:71) t ∈ U Sub( t ) = { t , . . . , t k } , (44)is satisfied. By Lemma 20 one finds that this relation is satisfied when U = MTree ( k ) .This completes the proof of Theorem 13. Recall that B ( N ) = (cid:70) (cid:53) k (cid:53) N B ( k, N ) is the set of nearest neighbor excluding indices.Given t ∈ T ree (Φ A ), take any V ⊂ Sub( t ) satisfying t ∈ V and Root( V ) ∈ B ( N ). Wesay that V is closely packed with respect to t when V = Sub o ( t ). Lemma 24 If V is not closely packed with respect to t , there exists S ∈ B ( N ) such that S ⊂ Root( t ) , | S | = | Root( V ) | and (cid:80) i ∈ S A i < (cid:80) i ∈ Root( V ) A i .ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice Proof.
Let n be the order of the nesting of the trees in Sub( t ). We prove the lemmaby induction on n . If n = 1 , V = Sub o ( t ) so there is nothing tobe proved. Suppose n ≥
3. Let V C = Sub( t ) \ V and a = min Root(Sub o ( t ) ∩ V C ).If such a does not exist, then V is closely packed. Suppose otherwise. We denote by s ∈ Sub o ( t ) ∩ V C the tree satisfying a ∈ Root( s ).(i) Suppose there exists u ∈ MSub( s ) such that both u ∈ V and Sub o ( u ) ∩ V (cid:54) = Sub o ( u )are satisfied. Then V u = V ∩ Sub( u ) is not closely packed with respect to u . Fromthe induction hypothesis, there exists S u ∈ B ( N ) such that S u ⊂ Root( u ) , | S u | = | Root( V u ) | and (cid:80) i ∈ S u A i < (cid:80) i ∈ Root( V u ) A i . Now the assertion of the lemma followson taking S = (Root( V ) \ Root( V u )) (cid:116) S u .(ii) Suppose otherwise, i. e. for any u ∈ MSub( s ) either u / ∈ V or Sub o ( u ) ⊂ V issatisfied.(a) Suppose Sub o ( u ) ⊂ V is satisfied for any u ∈ MSub( s ). This implies thatSub e ( s ) ⊂ V or equivalently Root e ( s ) ⊂ Root( V ). Let r ∈ Sub e ( t ) be the treethat directly straddles s , and q ∈ Sub o ( t ) be the one that directly straddles r .See Figure 7. We denote by P the branch point of q at which it straddles r .Note that Ht( P ) = Ht( r ). Let q be the subtree of q that extends downwardlyfrom P and is adjacent to r on its left side. By the definition of a we haveRoot o ( q ) ⊂ Root( V ). Define S ∈ B ( N ) as S = { Root( V ) \ (Root o ( q ) (cid:116) Root e ( s )) } (cid:116) (Root e ( q ) (cid:116) Root o ( s )) . (45)Then by Lemmas 22 and 23 we have (cid:80) i ∈ Root( V ) A i − (cid:80) i ∈ S A i ≥ Ht( P ) − Ht( s ) > s (cid:48) ∈ MSub( s ) such that s (cid:48) / ∈ V . Replace s by s (cid:48) and repeat the above arguments. Since the order of the nesting is finite,this case (ii)b cannot repeat endlessly, and we will eventually arrive at case (i)or (ii)a. The proof is completed. (cid:3) Given a state of the trop p-Toda A = ( A , . . . , A N ), there are generally more than one B ∗ ∈ B ( k, N ) which satisfy the condition min B ∈B ( k,N ) (cid:8)(cid:80) i ∈ B A i (cid:9) = (cid:80) i ∈ B ∗ A i . We wantto show that it is always possible to find such B ∗ that can be realized as the set of allthe bottom points of a forest , a set of trees in Φ A . Example 25
Consider Example 11 with k = 4 . One can take B ∗ = { , , , } , { , , , } , { , , , } or { , , , } . In Figure 5 one finds that the formertwo are not realized by forests, but the latter two are.ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice q q P r s a A Figure 7.
Nested trees q, r, s for the proof of Lemma 24, case (ii)a.
Lemma 26
For any ≤ k ≤ N there exists T ⊂ T ree (Φ A ) such that both Root( T ) ∈B ( k, N ) and min B ∈B ( k,N ) (cid:40)(cid:88) i ∈ B A i (cid:41) = (cid:88) i ∈ Root( T ) A i , (46) are satisfied.Proof. Recall that Φ A is a graph associated with A = ( A , . . . , A N ). It is composed ofseveral connected components called trees. We denote by N (Φ A ) the set of all nodes ofΦ A . It is the set of top points, bottom points, and branching points of the trees in Φ A .In the same way, we denote by L (Φ A ) the set of all links of Φ A . Note that, a top pointhas only a downward link, a bottom point has only an upward link, and a branchingpoint has an upward and several downward links outgoing from it.Choose any B ∗ ∈ B ( k, N ) that satisfiesmin B ∈B ( k,N ) (cid:40)(cid:88) i ∈ B A i (cid:41) = (cid:88) i ∈ B ∗ A i . (47)We draw a subgraph of Φ A associated with B ∗ , that is denoted by Φ A ( B ∗ ) and is definedas follows. First we adopt the bottom points { A i } i ∈ B ∗ as elements of N (Φ A ( B ∗ )), andadopt the links connected to them as those of L (Φ A ( B ∗ )). We also adopt the nodes atthe other end of these links as elements of N (Φ A ( B ∗ )). If such an adopted node is abranching point of Φ A , then there are two cases to be distinguished.(i) Filled branching point: all its downward links are adopted ones.(ii) Unfilled branching point: some of its downward links are unadopted ones. ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice L (Φ A ( B ∗ )) and N (Φ A ( B ∗ )). Repeat this procedure as muchas possible, and let Φ A ( B ∗ ) be the graph obtained finally. If all the branching pointsin Φ A ( B ∗ ) are filled ones, then there exists T ⊂ T ree (Φ A ) such that B ∗ = Root( T ).Hence we are done.Suppose otherwise. It is enough to show that there exists a procedure for findinga C ∗ ∈ B ( k, N ) such that the number of unfilled branching points in Φ A ( C ∗ ) is smallerthan the number of those in Φ A ( B ∗ ) by one, under the condition (cid:80) i ∈ C ∗ A i = (cid:80) i ∈ B ∗ A i .The claim of the lemma follows from using this procedure repeatedly. Now we startdescribing such a procedure. We denote by P an arbitrary chosen unfilled branchingpoint in Φ A ( B ∗ ) with lowest height, and by q ∈ Tree(Φ A ) the tree wherein P lies. Bydefinition, there are both adopted and unadopted subtrees of q under P . It is enoughto show that there exists a way to reduce the number of the adopted subtrees withoutchanging the other conditions.Among those subtrees, choose an adjacent pair of adopted/unadopted subtrees q , q . See Figure 8 for an example. Then, there is an unadopted tree r ∈ MSub( q )under P between q and q . By Lemma 24 and since there is no unfilled branchingpoint under P one sees that Sub( q ) ∩ Φ A ( B ∗ ) must be closely packed with respect to q , because otherwise (47) is not satisfied. P q q q r Figure 8.
An unfilled branching point P lies in the tree q . Thick lines are adoptedlinks, and thin lines are unadopted ones. Let B ∗ = (Root( q ) (cid:116) Root( r )) ∩ B ∗ . Then B ∗ = Root o ( q ) (cid:116) Root( V ) with some V ⊂ Sub( r ). We are to show that there exists a C ∗ ∈ B ( N ) satisfying | C ∗ | = | B ∗ | , (cid:88) i ∈ C ∗ A i = (cid:88) i ∈ B ∗ A i , (48)and can be written as C ∗ = Root( V (cid:48) ) with some V (cid:48) ⊂ Sub( q ) (cid:116) Sub( r ) ⊂ T ree (Φ A ). ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice s ∈ MSub( r ) such that both s ∈ V and Sub o ( s ) ∩ V (cid:54) = Sub o ( s )are satisfied. See Figure 9 (Left) for an example. Then V s = V ∩ Sub( s ) is notclosely packed with respect to s . By Lemma 24 one finds that this case cannothappen because otherwise (47) is not satisfied.(ii) Suppose otherwise, i. e. for any s ∈ MSub( r ) either s / ∈ V or Sub o ( s ) ⊂ V issatisfied.(a) Suppose Sub o ( s ) ⊂ V is satisfied for any s ∈ MSub( r ). See Figure 9 (Middle).This implies that Sub e ( r ) = V , hence B ∗ = Root o ( q ) (cid:116) Root e ( r ). Let C ∗ = Root e ( q ) (cid:116) Root o ( r ). Then | C ∗ | = | B ∗ | and by Lemmas 22, 23 wehave (cid:88) i ∈ C ∗ A i − (cid:88) i ∈ B ∗ A i ≤ Ht( r ) − Ht( P ) = 0 . (49)The inequality case is excluded because otherwise (47) is not satisfied.(b) Suppose otherwise, i. e. there exists r (cid:48) ∈ MSub( r ) such that r (cid:48) / ∈ V . SeeFigure 9 (Right). Replace r by r (cid:48) and repeat the above arguments. Since theorder of the nesting is finite, this case (ii)b cannot repeat endlessly, and we willeventually arrive at case (i) or (ii)a. The case (i) has already been excluded.The case (ii)a can be also excluded because now we have Ht( r (cid:48) ) − Ht( P ) < r r r s r Figure 9. (Left) case (i); (Middle) case (ii)a; (Right) case (ii)b
To summarize, the only possible case is (ii)a, under the condition that the equalityin (49) holds. Then by changing B ∗ by ( B ∗ \ B ∗ ) (cid:116) C ∗ one can reduce the number ofadopted subtrees under P by one without changing the other conditions. The proof iscompleted. (cid:3)
4. A continuous analogue of Kerov-Kirillov-Reshetikhin bijection
The Kerov-Kirillov-Reshetkhin (KKR) bijection is a bijection between the set of tensorproducts of crystals [21] and the set of a combinatorial objects known as rigged ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice sl case to explain the backgrounds of our algorithm in § L >
0, let P + = (cid:96) ∞ N =1 P + ,N where P + ,N = (cid:40) ( x , . . . , x N ) ∈ ( R > ) N (cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) i =1 x i ≤ L, k (cid:88) i =1 ( x i − − x i ) ≥ ≤ k ≤ N (cid:41) , (50)and M = (cid:96) ∞ s =1 M s where M s = (cid:40) λ = ( l j , m j ) ≤ j ≤ s ∈ ( R > × Z > ) s (cid:12)(cid:12)(cid:12)(cid:12) l > · · · > l s , s (cid:88) i =1 l i m i ≤ L/ (cid:41) . (51)Each element of the set M s is depicted as a Young diagram with area ≤ L/ λ ∈ M s define its j th vacancy number p j ( λ ) as p j ( λ ) = L − s (cid:88) k =1 min( l j , l k ) m k . (52)Also we define the set of quantum numbers or riggings Rig( λ ) associated with λ asRig( λ ) = (cid:26) ( J ( j ) i ) ≤ i ≤ m j , ≤ j ≤ s ∈ R m + ··· + m s (cid:12)(cid:12)(cid:12)(cid:12) ≤ J ( j )1 ≤ · · · ≤ J ( j ) m j ≤ p j ( λ ) for 1 ≤ j ≤ s (cid:27) . (53)Let Rig = (cid:96) λ ∈M Rig( λ ).Given L > φ : P + → M and φ : P + → Rig, such that φ = ( φ , φ ) gives a bijection φ : P + → (cid:70) λ ∈M { λ } × Rig( λ ). φ In order to adjust the values of the quantum numbers to conventional ones, we slightlymodify the algorithm in § • Suppose the last sequence of n ( i ) k ( i ) zeros is at the right end after a positive neighbor a as . . . , a, , . . . ,
0. Remove these zeros. If n ( i ) k ( i ) is odd then also remove a .We define the map φ : P + → M by the algorithm in § A -variables satisfying the highest weight condition, the Young diagram ˜ λ constructedby the algorithm in § § I , and the one in this section algorithm- II . Then wehave: Proposition 27
The Young diagram constructed by algorithm- I is equal to the one byalgorithm- II .Proof. To distinguish cases, we denote by x ( i ) ( X ) , y ( i ) ( X ) the sequences x ( i ) , y ( i ) constructed by algorithm- X (= I or II ). By induction on i , it is easy to see that both ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice x ( i )1 ( I ) ≥ x ( i )1 ( II ) , y ( i )1 ( I ) ≥ y ( i )1 ( II ), and x ( i ) j ( I ) = x ( i ) j ( II ) , y ( i ) j ( I ) = y ( i ) j ( II ) for j >
1. This implies that the set ofnumbers { ( µ ( i ) , ν ( i ) ) } ≤ i ≤ s constructed by both algorithms are the same. (cid:3) By this fact and Theorem 13, we see that φ is a map that yields the conservedquantities of trop p-Toda. Example 28
For the same state in Example 12 we have: x (1) = (4 , , , , , , , ,y (1) = (3 , , , , , , , ,x (2) = (3 , , , ,y (2) = (2 , , , ,x (3) = (2 , ,y (3) = (1 , , and µ (1) = µ (2) = µ (3) = 1 , ν (1) = 2 , ν (2) = ν (3) = 1 . The image of P + by the map φ is indeed in M . Lemma 29 φ ( P + ) ⊂ M .Proof. It suffices to show that the condition (cid:80) Ni =1 x i ≤ L in (50) leads to thecondition (cid:80) si =1 l i m i ≤ L/ y ( i ) j = x ( i ) j − µ ( i ) and (cid:80) N ( i +1) j =1 x ( i +1) j ≤ (cid:80) N ( i ) j =1 y ( i ) j as L ≥ (cid:80) N (1) j =1 x (1) j = 2 N (1) µ (1) + (cid:80) N (1) j =1 y (1) j ≥ (cid:80) si =1 N ( i ) µ ( i ) + (cid:80) N ( s ) j =1 y ( s ) j ≥ (cid:80) si =1 N ( i ) µ ( i ) = 2 (cid:80) si =1 l i m i . (cid:3) φ Consider the algorithm in § § i th block( µ ( i ) , ν ( i ) ) of the Young diagram ˜ λ we associate ν ( i ) non-negative real numbers calledquantum numbers. Recall that in the array of non-negative real numbers y ( i )1 , . . . , y ( i )2 N ( i ) we have k ( i ) sequences of zeros, and the j th sequence has n ( i ) j zeros. Denote by I ( i ) j theposition of the first element of the j th sequence. Let r ( i ) j = (cid:80) I ( i ) j − l =1 y ( i ) l . Then the ν ( i ) quantum numbers for the i th block are defined as r ( i )1 . . . r ( i )1 (cid:124) (cid:123)(cid:122) (cid:125) σ ( i )1 r ( i )2 . . . r ( i )2 (cid:124) (cid:123)(cid:122) (cid:125) σ ( i )2 . . . r ( i ) k ( i ) . . . r ( i ) k ( i ) (cid:124) (cid:123)(cid:122) (cid:125) σ ( i ) k ( i ) , (54)where σ ( i ) j = (cid:100) n ( i ) j / (cid:101) . Note that r ( i ) j < r ( i ) k for j < k . Given any positive real number L satisfying (cid:80) Nj =1 x i ≤ L , let λ ( j ) = (cid:80) jk =1 µ ( k ) and q j ( λ ) = L − s (cid:88) k =1 min( λ ( j ) , λ ( k ) ) ν ( k ) , (55) ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice ≤ j ≤ s . By Theorem 13, we have λ ( s +1 − j ) = l j , ν ( s +1 − j ) = m j , and q s +1 − j ( λ ) = p j ( λ ).By the following lemma, one sees that the quantum numbers for the i th block obeythe condition 0 ≤ r ( i ) j ≤ q i ( λ ). This enables us to define the map φ : P + → Rig bythe procedure described above and by identifying the m s +1 − i = ν ( i ) quantum numbersin (54) with J ( s +1 − i )1 , . . . , J ( s +1 − i ) m s +1 − i for 1 ≤ i ≤ s in (53). Lemma 30 φ ( P + ) ⊂ Rig .Proof.
By definition, the relation 0 ≤ r ( i ) j holds trivially. Let L = L − (cid:80) Nj =1 x i . Itsuffices to show that the relation (cid:80) N ( i ) j =1 y ( i ) j ≤ q i ( λ ) − L is satisfied for 1 ≤ i ≤ s byinduction on i , since then we have r ( i ) j ≤ (cid:80) N ( i ) j =1 y ( i ) j ≤ q i ( λ ). It is done by using therelations in the proof of Lemma 29 as well as the relation q j ( λ ) = q j − ( λ ) − µ ( j ) N ( j ) where q ( λ ) = L . (cid:3) Example 31
For the A -variables in Example 11 the quantum numbers for the first blockof the Young diagram are √ − , √ − , √ − , those for the second are , , andthat for the third is √ − .4.4. The inverse map Having obtained the pair φ = ( φ , φ ) that gives a map φ : P + → (cid:70) λ ∈M { λ } × Rig( λ ),now we consider its inverse map φ − . This is done by using the inverse of thealgorithm in § § λ = ( l j , m j ) ≤ j ≤ s ∈ M s and J = ( J ( j ) i ) ≤ i ≤ m j , ≤ j ≤ s ∈ Rig( λ ). By the correspondence in the previous subsection,we regard λ as λ = { ( µ ( i ) , ν ( i ) ) } ≤ i ≤ s , and J as J = { ( r ( i ) j ) σ ( i ) j } ≤ j ≤ k ( i ) , ≤ i ≤ s ∈ Rig( λ )where σ ( i ) j is the multiplicity of r ( i ) j . The quantum numbers obey the condition0 ≤ r ( i )1 < · · · < r ( i ) k ( i ) ≤ q i ( λ ) where q i ( λ ) is the vacancy number defined by (55).Given ( λ, J ), its image x (1) = ( x (1)1 , . . . , x (1)2 N (1) ) under the map φ − is given by such astep-by-step construction as y ( s ) → x ( s ) → y ( s − → x ( s − → · · · → y (1) → x (1) . Thefirst step goes as follows. Let r ( s )1 . . . r ( s )1 (cid:124) (cid:123)(cid:122) (cid:125) σ ( s )1 r ( s )2 . . . r ( s )2 (cid:124) (cid:123)(cid:122) (cid:125) σ ( s )2 . . . r ( s ) k ( s ) . . . r ( s ) k ( s ) (cid:124) (cid:123)(cid:122) (cid:125) σ ( s ) k ( s ) be the ν ( s ) quantum numbers for the s -th (top) block. We define a sequence of non-negative real numbers y ( s )1 , . . . , y ( s )2 N ( s ) as r ( s )1 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( s )1 − , r ( s )2 − r ( s )1 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( s )2 − , . . . , r ( s ) k ( s ) − r ( s ) k ( s ) − , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( s ) k ( s ) − . Then we define a sequence of positive real numbers x ( s )1 , . . . , x ( s )2 N ( s ) by x ( s ) j = y ( s ) j + µ ( s ) .The subsequent steps go as follows. Given x ( i +1)1 , . . . , x ( i +1)2 N ( i +1) and the quantumnumbers in (54), we define a sequence of non-negative real numbers y ( i )1 , . . . , y ( i )2 N ( i ) by ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice w ( i +1) k = (cid:80) kj =1 x ( i +1) j for 1 ≤ k ≤ N ( i +1) and w ( i +1)0 = 0. For each r ( i ) j either w ( i +1) k ≤ r ( i ) j < w ( i +1) k +1 for some 0 ≤ k ≤ N ( i +1) − r ( i ) j ≥ w ( i +1)2 N ( i +1) is satisfied.Roughly speaking, we split x ( i +1) k +1 and insert some zeros in the former case, while in thethe latter case we append r ( i ) j − w ( i +1)2 N ( i +1) and some zeros at the end of the sequence.To be more precise, let us consider the case w ( i +1) k ≤ r ( i ) j < r ( i ) j +1 < w ( i +1) k +1 and the case r ( i ) k ( i ) > r ( i ) k ( i ) − ≥ w ( i +1)2 N ( i +1) as examples, where we assumed no other r ( i ) l ’s exist in the (half-)intervals determined by w ( i +1) k ’s. In the former case we replace x ( i +1) k +1 = w ( i +1) k +1 − w ( i +1) k by r ( i ) j − w ( i +1) k , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( i ) j − , r ( i ) j +1 − r ( i ) j , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( i ) j +1 − , w ( i +1) k +1 − r ( i ) j +1 . In the latter case we add the following sequence after x ( i +1)2 N ( i +1) : r ( i ) k ( i ) − − w ( i +1)2 N ( i +1) , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( i ) k ( i ) − − , r ( i ) k ( i ) − r ( i ) k ( i ) − , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) σ ( i ) k ( i ) − . It is easy to generalize these procedures for the cases where any number of differentvalues of the quantum numbers exist in the (half-)intervals determined by w ( i +1) k ’s .Then we define a sequence of positive real numbers x ( i )1 , . . . , x ( i )2 N ( i ) by x ( i ) j = y ( i ) j + µ ( i ) .Given λ = { ( µ ( i ) , ν ( i ) ) } ≤ i ≤ s ∈ M s and J = { ( r ( i ) j ) σ ( i ) j } ≤ j ≤ k ( i ) , ≤ i ≤ s ∈ Rig( λ ) , wedefine the map φ − by φ − ( λ, J ) = ( x (1)1 , . . . , x (1)2 N (1) ). By construction, we see that it isindeed the inverse of the map φ . Moreover we have the following: Lemma 32 φ − ( (cid:70) λ ∈M { λ } × Rig( λ )) ⊂ P + .Proof. From Lemma 10 we see that the above algorithm for φ − preserves the highestweight condition. Hence it suffices to show that w (1)2 N (1) := (cid:80) N (1) j =1 x (1) j ≤ L . To beginwith, we prove (cid:80) N ( i ) j =1 y ( i ) j ≤ q i ( λ ) for 1 ≤ i ≤ s . For i = s , it is satisfied as (cid:80) N ( s ) j =1 y ( s ) j = r ( s ) k ( s ) ≤ q s ( λ ). Suppose (cid:80) N ( i +1) j =1 y ( i +1) j ≤ q i +1 ( λ ) for some i < s . Thenwe have w ( i +1)2 N ( i +1) = (cid:80) N ( i +1) j =1 y ( i +1) j + 2 N ( i +1) µ ( i +1) ≤ q i +1 ( λ ) + 2 N ( i +1) µ ( i +1) = q i ( λ ),and hence (cid:80) N ( i ) j =1 y ( i ) j = max { w ( i +1)2 N ( i +1) , r ( i ) k ( i ) } ≤ q i ( λ ). Thus by descending inductionon i this inequality holds for any 1 ≤ i ≤ s . Now we obtain the desired result as w (1)2 N (1) = (cid:80) N (1) j =1 y (1) j + 2 N (1) µ (1) ≤ q ( λ ) + 2 N (1) µ (1) = q ( λ ) = L . (cid:3) By Lemmas 29, 30 and 32 we obtain the following result.
Theorem 33
The map φ : P + → (cid:70) λ ∈M { λ } × Rig( λ ) is a bijection. So far we do not know whether one can regard this bijection as an isomorphism, i. e. wedo not know what kind of mathematical structures are preserved under this bijection.
5. Concluding remarks
In this paper we elucidated combinatorial aspects of the conserved quantities of generaltropical periodic Toda lattice beyond the generic condition. Let us summarize what we ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice § × ’s appear in a given level. In the 10-eliminationalgorithm for pBBS this corresponds to simultaneous disappearance of more than twoconsecutive blocks, for which the rule to draw lines was ambiguous in [16]. (ii) Weformulated our original lemmas in § § any B ∗ in (47) must be realizedby a forest, on which their proof substantially depend. But as we have shown in Example25 this claim is not true. The correct statement is that at least one B ∗ in (47) can berealized by a forest, as we proved in § sl case [17] the set P + is a subset ofthe tensor product of crystals B ⊗ L with B = { , } and its elements are expressed as1 . . . (cid:124) (cid:123)(cid:122) (cid:125) x . . . (cid:124) (cid:123)(cid:122) (cid:125) x . . . . . . (cid:124) (cid:123)(cid:122) (cid:125) x N − . . . (cid:124) (cid:123)(cid:122) (cid:125) x N . . . (cid:124) (cid:123)(cid:122) (cid:125) x N +1 for some N with the condition in (50), where we omitted ⊗ symbols. In our algorithmthe integer x i ’s here have been replaced by continuous variables taking their values in realnumbers. Actually our algorithm is based on the algorithm in [14] which is a variation ofthe original algorithm. In sl n +1 case the set B is replaced by B = { , . . . , n + 1 } and thehighest weight condition is adequately modified. The algorithm of the KKR bijectionfor sl n +1 case was given in § sl case has been developed yet. Therefore a promising way to construct acontinuous analogue of the KKR bijection for sl n +1 is to develop such a variation first.Then the remaining task will be rather straightforward.Finally we would like to explain the difference of the arguments between [17] and thepresent work. In [17] the conservation of the Young diagram under the time evolutionof pBBS was shown by the following way. For any positive integer l and any state p , we introduced a time evolution T l ( p ) and an energy E l ( p ) (Proposition 2.1) byusing the crystal theory and its energy function. Then the conservation of the energy E l ( T k ( p )) = E l ( p ) and the commutativity of the time evolutions T l T k ( p ) = T k T l ( p )were shown (Theorem 2.2). Finally we proved that the data carried by the whole setof the values of the energy E l ( p ) ( l = 1 , , . . . ) was equivalent to the Young diagram ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice E l ( p ) can be allocated in the present paper. However, the authorthinks that one can generalize the above argument to trop p-Toda case because thearguments in [4] to construct a commuting family of time evolutions may be used todevelop an analogue of the energy function in this case. We hope to report any progresson this subject in the near future. Acknowledgement . This work was supported by JSPS KAKENHI Grant Number25400122. [1] Toda M 1981
Theory of nonlinear lattices (Springer-Verlag: Berlin-New York)[2] Athorne C, Maclagan D and Strachan I (ed) 2012
Tropical Geometry and Integrable Systems (Contemporary Math ) (Providence, Rhode Island: American Mathematical Society)[3] Inoue R, Kuniba A and Takagi T 2012 Integrable structure of box-ball systems: crystal, Betheansatz, ultradiscretization and tropical geometry, J. Phys. A , 073001 (63 pages)[4] Takagi T 2012 Commuting time evolutions in the tropical periodic Toda lattice, J. Phys. Soc. Jpn. , 104005 (7 pages)[5] Inoue R and Iwao S 2012 Tropical curves and integrable piecewise linear maps, ContemporaryMath. , 21–39[6] Kimijima T and Tokihiro T 2002 Initial-value problem of the discrete periodic Toda equations andits ultradiscretization, Inverse Problems , 995–1021[9] Berenstein A and Kazhdan D 2000 Geometric and unipotent crystals, in Visions in Mathematics,GAFA 2000 Special Volume, Part I, eds. N. Alon, J. Bourgain, A. Connes, M. Gromov, andV. Milman, (Birkh¨auser: Basel), 188–236[10] Itenberg I, Mikhalkin G and Shustin E 2009
Tropical Algebraic Geometry (Birkh¨auser: Basel)[11] Maclagan D 2012 Introduction to tropical algebraic geometry, Contemporary Math. , 1–19[12] Kerov S V, Kirillov A N and Reshetikhin N Yu 1988 Combinatorics, the Bethe ansatz andrepresentations of the symmetric group. J. Soviet Math. , 12987–13021[17] Kuniba A Takagi T and Takenouchi A 2006 Bethe ansatz and inverse scattering transform in aperiodic box-ball system Nucl. Phys. B [PM] 354–397[18] Takagi T 2010 Level set structure of an integrable cellular automaton, SIGMA, 6, 027, 18 pages[19] Arnold V I 1989 Mathematical Methods of Classical Mechanics (New York: Springer-Verlag) 2nded, p. 287[20] Tokihiro T 2010
Mathematics of box-ball systems (Tokyo: Asakura Shoten) [in Japanese][21] Kashiwara M 1991 On crystal bases of q -analogue of universal enveloping algebras, Duke Math. J. ombinatorial aspects of the conserved quantities of the tropical periodic Toda lattice63