Tropical integrable systems and Young tableaux: Shape equivalence and Littlewood-Richardson correspondence

We present a new characterization of the shape equivalent class and the Littlewood-Richardson correspondence of Young tableaux in terms of tropical (ultradiscrete) integrable systems. As an application, an alternative proof of the"shape change theorem"is given.


Introduction
The tropicalization is the transformation through which the structure of rings (+, ×, −1 ) is transformed into the structure of semi-fields (min, +, −). For example, the tropicalization of a polynomial function is a piecewise linear function. It is known there are a lot of interesting examples in the field of combinatorics, mathematical physics, etc. where the tropicalization provides a new insight and an application. One of the most significant case is the Young tableau. The earliest study on the tropical aspects of Young tableaux was made by A. N. Kirillov [5], who introduced the geometric RSK correspondence (originally, tropical RSK correspondence 1 ). This correspondence was studied further by M. Noumi and Y. Yamada [8] by means of tropical (ultradiscrete) integrable systems. In [8], they showed the fact that the row-bumming algorithm is expressed as a recurrence equation of tropical matrices, which is the tropicalization of the discrete Toda equation. This technique has been accepted as a fundamental tool for studies on the combinatorics of Young tableau and related topics. (For resent studies on the geometric RKS correspondence and its applications, see [1,7,9] and references therein.) On the other hand, Y. Mikami [6], Y. Katayama, and S. Kakei [4] introduced another new correspondence between Young tableaux and tropical integrable systems. Interestingly enough, their correspondence is apparently independent of Noumi and Yamada's correspondence. In 2018, Iwao [3] presented a tropical characterization of the rectification of skew tableaux based on these correspondences. We would expect that the combination of Noumi-Yamada's geometric tableaux and Katayama-Kakei's correspondence provides a rich tropical interpretations of the combinatorics of Young tableaux.
Date: March 19, 2018. 1 The word "tropical" nowadays has a different meaning. Many researchers prefer to use the "geometric RSK correspondence" instead. This work is a continuation of [3]. In Section 2, we briefly review the previous work [3]. The jeu de taquin slide, which is a fundamental procedure of the combinatorics of Young tableaux, is expressed by the recurrence formula (3). This formula is equivalent to the combinatorial procedure ϕ k ( §2.3). The rectification of Young tableaux is now expressed as a composition of finitely many ϕ k 's (see (9)). It is diagrammatically expressed by a planar diagram such as (9). From this diagram, one can induce some horizontal diagram (see (10)) by getting each column "together in one bundle." It is showed that these horizontal diagrams can be defined independently of the choice of planer diagrams.
In Section 3, we deal with the "dual" object to the previous section. It would be natural to ask "what will happen if one gets each row of the planer diagram together in one bundle?" The result is some vertical diagram (see (14)). We show (Proposition 3.5) that this vertical diagram is determined independently of the choice of planer diagrams. The proof is based on the fact that the diagram is characterized by some L-formula ( §1.2). As an application, we give a proof of the "shape change theorem [2]" (Theorem 3.11).
One can do both procedures for rows and columns simultaneously. In Section 4, we introduce some "concentrated" diagram by getting each row and each column together in one bundle (see (15)). It is shown that this diagram is closely related with the Littlewood-Richardson correspondence. Moreover, we present a new characterization of the Littlewood-Richardson correspondence in terms of the tropical mathematics (Theorem 4.2).
For convenience of readers, we gave a short introduction in a combinatorics of Young tableaux in §A. The definition of the Takahashi-Satsuma Box-Ball system is given in §B.

1.1.
Notations for Young tableaux. In this article, we follow the conventions in Fulton's book [2]. Let λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ ) be a Young diagram. A semistandard tableau of shape λ is obtained by filling the boxes in λ with a number according to the following rules: (i) in each row, the numbers are weakly increasing from left to right, (ii) in each column, the numbers are strongly increasing from top to bottom. A semi-standard tableau is often referred to as tableau shortly. A tableau with n boxes is called standard if it contains distinct n numbers 1, 2, . . . , n. Let λ/µ be a skew diagram. A skew (semi-standard) tableau of shape λ/µ is obtained by filling the boxes with a number according to the same rule for a tableau. If a skew tableau with n boxes contains distinct n numbers 1, . . . , n, it is said to be standard. See §A for other definitions.

1.2.
Notations for tropicalization. We use the notations of mathematical logic in order to simplify arguments for tropicalization. For details, see [3].
The word "L-term" means a subtraction-free rational function. The morphism The correspondence among combinatorial, tropical, and geometric objects is given as follows:

Tropical (ultradiscrete) KP and jeu de taquin
In this section, we shortly review the previous results [3]. For definitions of the terms "jeu de taquin slide," "inside corner," "outside corner,"etc, see §A.
The following theorem is due to [4]: 6,4]). For a sequence of skew tableaux S 0 , S 1 , S 2 , . . . , set F t i,j = the number of boxes in 1 st , 2 nd , . . . , i th rows in S t which are indexed by a number smaller than or equal to j , where an empty box is considered as a box indexed by 0. If each S t+1 is obtained from S t by a jeu de taquin slide, (F t i,j ) i,j,t satisfies the tropical KP equation (1). Proposition 2.2 ([3]). By setting
By definition of skew tableaux, W i,j must be non negative. Moreover, W i,j satisfies the following conditions: There exists some d such that i > d ⇒ W i,j = 0 for all j.
Consider the mapping W : Ω → X which corresponds a skew tableau with the matrix (W i,j ) i,j .

Proposition 2.3 ([3]). W is bijective.
We always identify Ω with X. While the matrix W is of infinite size, it is "essentially finite" because of (4) and (5). To simplify notations, we often regard W as a matrix of size d × N .
2.3. Jeu de taquin slide starting from the k th row. Now we construct the map ϕ k : X → X for any positive integer k, which is a tropical counterpart of the jeu de taquin slide starting from k th row. Let W = (W i,j ) ∈ X and [k] = (0, . . . , 0, k ∨ 1, 0, . . . ). The definition of ϕ k is given as follows [3]: (2) Compute Q i,j , W + i,j recursively as follows: If the vector Q j = (Q 1,j , Q 2,j , . . . ) is already defined for some j ∈ Z ≥0 , define the new vectors Q j+1 = (Q 1,j+1 , Q 2,j+1 , . . . ) and W + j = (W + 1,j , W + 2,j , . . . ) by the formula . We regard (7) as a recurrence formula whose inputs are Q j and W j , and outputs are Q j+1 and W + j . It is convenient to draw a diagram such as the inputs are written on the left and top sides, and the outputs are on the right and bottom sides. The whole procedure to calculate ϕ k (W ) is diagrammatically displayed as Moreover, the map ϕ k also admits a combinatorial interpretation as follows: • Draw a path on the matrix W = (W i,j ) by the following rule (see §2.4): -The path starts at the (k, 0) th position.
-When the path reaches at the (i, j) th position, extend it to the lower right neighbor if W i,j = 0, or to the right neighbor if W i,j = 0. • For each non-zero number W i,j on the path, decrease it by one and increase the number at the upper neighbor by one; The matrix given by this procedure coincides with ϕ k (W ). 2.5. Rectification. Any skew tableau reaches a (non-skew) tableau thorough a sequence of finitely many jeu de taquin slides. To repeat jeu de taquin slides is nothing but to choose inside corners repeatedly. By filling numbers in chosen inside corners in decreasing order, one obtains a standard tableau. For example, if we apply the sequence of jeu de taquin slides to We call the (non-skew) tableau obtained by this procedure the rectified tableau. With diagrammatic expressions as in §2.3, this procedure is displayed as If [k] is placed on the leftmost column, it represents the jeu de taquin slide starting from the k th row The diagram (9)  2.6. Associated tableaux. For diagrams such as (9), we assign the tableau t 1 ← t 2 ← . . . ← t d to a column whose entries are [t 1 ], [t 2 ], . . . , [t d ] from top to bottom. For example, from (9), we induce the new diagram (10) (1, 1, 0) (1, 0, 0) (1, 0, 2) (1, 0, 2) (1, 1, 3)· · · 1 1 1 2 It was proved by Iwao [3] that if one exchanges the entries in the leftmost column in (9) gives the same tableau, one also obtain the same diagram (10). This means that the diagram (10) is well-defined independently of (9). The tableau associated with each column is called the associated tableau.   2.7. Lift to M. All procedures in §2.1- §2.6 can be expressed in terms of the language L and the L-structure M. They can be lifted to M appropriately, and their "geometric" counterparts are expressed as a discrete integrable system.

For example, let
We can uniquely define the rational map , which is a discrete analog of the Toda equation. It is verified that its tropicalization coincides with (3), where Q t i,j = I t i,j and W t i,j = V t i,j are the tropical variables. All other procedures and facts can also be lifted to M and expressed in terms of the discrete integrable systems. See [3] for details. Note that the shapes of these two sequence coincides with each other. In such case, two skew tableaux are said to have the same shape changes by 1 2 3 4 . The following theorem is referred to as the "shape change theorem [2, Appendix A]." Theorem 3.1 (Shape change theorem). If two skew tableaux have the same shape changes by some standard tableau, then they actually have the same shape changes by any standard tableau.
If two skew tableaux have the same shape changes by some standard tableau (therefore, if they have the same shape changes by any standard tableau), they are said to be shape equivalent.
Moreover, for a real infinite vector V , define the matrix F k (V ) by Consider the map M ∞ → M ∞ ; (V i,j ) → (U i,j ) uniquely defined by the equation By induction on N , we can verify that this map is expressed by L-terms ( §1.2).
Remark 3.3. This map can be regarded as an "F -matrix version" of Noumi-Yamada's geometric tableau. In fact, the geometric tableau (I i,j ) → (J i,j ) is defined by the equation where E k (J) is a matrix analogously defined to F k (V ). For details, see [3, Section 5].
Let W i,j = V i,j and L i,j = U i,j be the tropical variables. Through the tropicalization, we obtain the piecewise linear map map has a interesting combinatorial interpretation which we explain below. A circled array is a collection of finitely many rows that consists of circled numbers, where the numbers are arranged in increasing ordering. (Empty rows are allowed.) For example, denote the circled array given by acting the one-rowed arrays x 1 , x 2 , x 3 , . . . to the empty array.
Theorem 3.4. Let W j = (W 1,j , W 2,j , . . . ) be a sequence of nonnegative integers that satisfies i W i,j < ∞. Let w j denote the one-rowed array that consists of W 1,j 1's, W 2,j 2's,. . . . Then the map (W i,j ) → (L i,j ) has the following combinatorial interpretation: The number of i's in the j th row of the circled array Here N is a sufficiently large integer.
Proof. Let V, V ′ , U, U ′ ∈ M ∞ be vectors of infinite length. For any k, there uniquely exists the map . This equation is noting but the Lax formulation of the discrete Toda equation (see, for example, ). It is well known that its tropical counterpart is the Takahashi-Satsuma box-ball system. In this context, the action w a ↼ w b can be seen as the tropical counterpart of the equation and U (i) associated with the i th row of the circled array. Generally, the action (w a1 ↼ w a2 ↼ · · · w aj ) ↼ w b is the counterpart of This proves the relation between (12) and the circled array.
. It is also verified that this map can be expressed by L-terms. The tropicalization Proof. It is enough to prove the existence and uniqueness of the map M ∞ → M ∞ ; (U, I) → (U ′ , I ′ ) such that F k (U ′ )E(I) = E(I ′ )F k (U ) for any k, which is easily verified by straightforward calculation. Obviously, this map is expressed by L-terms.

Let us introduce the new variables
We write this map diagrammatically as .
From Theorem 3.4, the data (L 1 , . . . , L N +1 ) naturally corresponds with a circled array, which we will call the associated circled array. Because the data (L 1 , . . . , L N +1 ) is determined by a skew tableau (≃ an array W ), there exists a natural correspondence from the set of skew tableaux to the set of circled arrays.
Example 3.6. The skew tableau To simplify notations, we often omit to write empty rows at the bottom of an array. We let the sign "∅" denote the array that consists of empty rows.

3.4.
Proof of Shape change theorem. We often look the diagrams "upsidedown." In other words, we look the data at right and bottom as inputs and at top and left as outputs. Combinatorially, this corresponds with the reverse slide [2, §1.2]. As its name suggests, the reverse slide is the reverse operation of the jeu de taquin slide.
Lemma 3.7. The associated circled array of any (non-skew) tableau is ∅.
Proof. Let W = (W i,j ) be the matrix associated with a (non-skew) tableau, and w j be the one-rowed array associated with the vector W j = (W 1,j , W 2,j , . . . ).
Since any tableaux contains no empty box, the relation i > j ⇒ W i,j = 0 holds. Especially, the array w j consists of numbers lower than or equal to j. For sufficiently large N , one can verify by backward induction on j ≤ N that all numbers in the circled array w N ↼ w N −1 ↼ · · · ↼ w j are lower than or equal to j. Hence the circled array w N ↼ w N −1 ↼ · · · ↼ w 0 consists of empty rows. When we see the diagram "upside-down," the sequence a 1 , . . . , a k and ∅ (see Lemma 3.7) are thought of the inputs. This implies that b 1 , . . . , b k depends only on a 1 , . . . , a k . Example 3.9. When a sequence of reverse slides starting from (2, 1, 2, 4, 3) th rows operated to any tableau, empty boxes must arise in (1, 1, 2, 3, 2) th rows. For example, see . Corollary 3.10. The associated circled array of a skew tableau is determined by the sequence a 1 , a 2 , . . . , a k .
Proof. It suffices to see the diagram (14).
Then we have the following theorem which contains the shape change theorem: Theorem 3.11. For two skew tableaux of the same shape, the following (a-c) are equivalent: (a) They are shape equivalent. (b) They have the same shape changes by some standard tableau. (c) Their associated circled arrays coincide with each other.
Proof. (a)⇒(b) is obvious. For (b)⇒(c), assume that two skew tableaux admit a sequence of jeu de taquin slides starting from b th 1 , b th 2 , . . . , b th k rows, and reach to tableaux where outside corners in a th 1 , a th 2 , . . . , a th k rows has been changed. Since their associated circled arrays depend on a i 's only (Corollary 3.10), they coincide with each other. For (c)⇒(a), consider the diagram such as (14). Since the circled arrays at the top are same, the changes that occur outside corners by the sequence of jeu de taquin also coincide. From Theorem 3.11, it can be said that the shape equivalent class of a skew tableau is completely determined by its associated circled array. . Remark 3.13. There exists an alternative way to determine the shape equivalent class by calculating the "Q-tableau" [2,§AppendixA]. This can be calculated easily, but essentially depends on the shape of the skew tableau. The circled array, however, depends only on the order of reverse slides (Corollary 3.10). It would be said that the circled array presents "universal" information of the shape equivalent class.

Application 2: Littlewood-Richardson correspondence
As we have seen in the previous sections, the diagram of rectification (9) induces a new useful diagram if one gets its rows or columns together in one bundle. If we do both procedures, we obtain the following diagram: .
This diagram defines the one-to-one correspondence (L i , P i ) ↔ (L ′ i , P ′ i ). For example, from the diagram (14), we have 2 1 1 1 In general, for any two Young diagrams µ ⊂ λ, we have where M is a circled array and Z is a tableau of shape µ. We will see that the correspondence is related closely with the Littlewood-Richardson correspondence [2, §A.1].

4.1.
Definition of the Littlewood-Richardson correspondence. We review the definition of the Littlewood-Richardson correspondence. For the definitions of the terms RSK correspondence and P tableau, see the standard textbook [2].
Similar to the case of jeu de taquin slides, repeating reverse slides is nothing but choosing outside corners repeatedly. In other words, this is equivalent to specify a standard skew tableau "sticking" outside of the tableau. For example, the sequence of reverse slides to There exists a useful correspondence between sequences of reverse slides and tableaux pairs. Here we give its outline briefly according to Fulton [2,§Appendix A].
Let R be the standard skew tableau sticking outside of the tableau X. For arbitrary tableau V • of shape λ, one can find a word w = t 1 t 2 . . . t m (m = |µ|) and a tableau U of shape µ such that • when t m−i is column-bumped, the new box evokes at the place of i by "reverse column bumping algorithm." Let T = P (w) denote the P -tableau associated with w. We call the pair [T, U ] the Littlewood-Richardson pair.
Let S(λ/µ, X) be the set of skew tableaux of shape λ/µ, the rectification of which is X. On the other hand, let T (λ, µ, V • ) denote the set of the pairs [T, U ] where T is of shape λ, U is of shape is µ, and T · U = V • . Let be the map that associates the skew tableau S, given from X through a sequence of reverse slides defined by R, with the Littlewood-Richardson pair [T, U ]. It is known that this map is bijective [2, §5.1, Proposition 2]. This map is called the Littlewood-Richardson rule. Moreover, for arbitrary two tableaux X, Y of the same shape, one can define the bijection This bijection S(λ/µ, X) ↔ S(λ/µ, Y ) is called the Littlewood-Richardson correspondence by V • . Later we will see that it is independent of the choice of V • .

Tropical interpretation of Littlewood-Richardson correspondence.
As we have seen in the previous subsection, the Littlewood-Richardson correspondence is essentially defined via the standard skew tableau R. Let R(λ, X) be the set of standard skew tableaux R sticking outside of X, where the outside of R is of shape λ. Therefore, we have the commutative diagram: where f is the reverse column bumping and g is a composition of f and the Littlewood-Richardson rule.
Note that the equation f (R) = S is equivalent to the diagram where the matrices W = (W j ) and W ′ = (W ′ j ) correspond with S and X respectively. By introducing the equivalence relation R 1 ∼ R 2 ⇐⇒ the associated tableaux of R 1 and R 2 are same over R(λ, X), we induce the two one-to-one correspondences (R(λ, X)/ ∼) ↔ S(λ/µ, X), (R(λ, X)/ ∼) ↔ T (λ, µ, V • ).
Because the shapes of U (µ) and Z are same, each element of R(λ, X)/ ∼ can be identified with its associated tableau of shape µ. If T (µ) denotes the set of tableaux of shape µ, we finally obtain the maps   Acknowledgment. This work is partially supported by JSPS KAKENHI:26800062.
Appendix A. Basics on the combinatorics of Young tableaux A box B in a Young diagram is said to be placed in the corner if there exists no box below nor on the right of B. For a skew diagram λ/µ, the corner of λ is called the outside corner and the corner of µ is called the inside corner.
For a skew tableau T and an inside corner b, the jeu de taquin starting from b is defined as follows: (i) Compare the two numbers in the boxes below and on the right of the hole b, and slide a box with smaller number to b. If these two numbers are same, slide the box below b. (ii) Compare the two numbers in the boxes below and on the right of the hole, and slide a box according to the same rule in (i). (iii) Repeat (ii) until the hole reaches to the outside corner. as the following combinatorial rule: (i) Create a copy of each ball. (ii) Move each copy to the nearest empty box on the right. (iii) Delete the original balls. This procedure is called the time evolution role of the Takahashi-Satsuma Box-Ball system. The following is an example of the combinatorial expression of the time · · · · · · (L ′ i , W ′ i ); · · · · · · For more detail, see