Séminaire Lotharingien de Combinatoire 59 (2010), Article B59g

ACTIONS OF THE SYMMETRIC GROUP GENERATED BY COMPARABLE SETS OF INTEGERS AND SMITH INVARIANTS

OLGA AZENHAS AND RICARDO MAMEDE

Abstract. Lascoux and Schützenberger have shown that there exists a unique action of the symmetric group generated by the commutation of the column lengths of a twocolumn tableau and preserving the plactic class. We describe more general operators on pairs of comparable subsets of {1, …, n} which commute their cardinalities, and we prove that those operators define an action of the symmetric group by checking the braid relations on triples of sets of integers. The action of the symmetric group by Lascoux and Schützenberger appears in our construction as an extreme case as we only require the invariance of the shape and the weight of the insertion tableau. Instead of sets of positive integers one may take other equivalent objects as words in a two letter alphabet, and describe an action of the symmetric group on words congruent to key-tableaux defined by reflection crystal operators type based on nonstandard pairing of parentheses. This construction arises naturally as a combinatorial description of the Smith invariants of certain sequences of products of matrices, over a local principal ideal domain, under a natural action of the symmetric group.

1. Introduction

In [20, 21] (see also [26]), A. Lascoux and M.-P. Schützenberger introduced the concept of key-tableau as a Young tableau whose columns are comparable under the inclusion order, and they have used this notion to study Demazure characters combinatorially, see [7]. For type A, Demazure characters are equivalent to key polynomials. Due to the action of the symmetric group on the set of tableaux, originally defined by Ehresmann in [8], the symmetric group acts on the set of key-tableaux. This action coincides with the one defined by the reflection crystal operators on the free algebra, based on the standard matching of parentheses, a particular parentheses matching on words in a two-letter alphabet. Reflection crystal operators are due to Lascoux and Schützenberger [19, 23], and they are equivalent to the ones coming from the theory of crystal graphs in the work of Kashiwara and Nakashima [24].

A column is a word in which every letter is strictly less than the previous one. Every word w can be uniquely written as a product of a minimal number of columns, and the sequence formed by their lengths is called the column-shape of w. A word w is frank if its shape is a permutation of the sequence formed by the column lengths of the only tableau in the Knuth class of w [21]. The action of the symmetric group on words congruent to key-tableaux defined by the reflection crystal operators preserve

2000 Mathematics Subject Classification. 05E05, 05E10, 15A45. Key words and phrases. Action of the symmetric group; Young tableaux; frank words; jeu de

taquin; pairing of parentheses; reflection crystal operators; Smith invariants.

The first author was supported by CMUC/FCT and SFRH/BSAB/515/2005, grant by the Portuguese Foundation of Science and Technology, FCT, at the Combinatorics Group, Fakultät für Mathematik, Universität Wien, Austria, and the second by CMUC/FCT.

the Q-symbol, and has a translation on frank words within a Knuth class defined by jeu de taquin slides on two-column frank words [21]. In this paper, we drop the requirements of preserving respectively the Q-symbol and the Knuth class in those two constructions, and we define two families of actions of the symmetric group including, as particular cases, the actions of the symmetric group described above. The first one over frank words runs over tableaux of the same shape and weight, and the second over words congruent to key-tableaux. More specifically, we consider the ordered set P [n] of sets of positive integers in [n] = {1, …, n} and describe operators on pairs of comparable elements of P [n] which generalize to non-congruent frank words the action of the symmetric group defined by jeu de taquin slides on two-column frank words within a Knuth class. This action of the symmetric group has a translation on words congruent to key-tableaux defined by reflection crystal operators type based on nonstandard pairing of parentheses already considered in [6]. Such a translation is described by the dual Robinson–Schensted–Knuth correspondence [9, 17]. (See also [22].)

These two families of actions of the symmetric group arise naturally in the matrix context. Given an n by n non-singular matrix A, with entries in a local principal ideal domain with prime p, we write A ∼ Δ_{α} to mean that by Gaußian elimination one can reduce A to a diagonal matrix Δ_{α} with diagonal entries p^{α_{1}}, …, p^{α_{n}}, for unique non-negative integers α_{1} ≥ … ≥ α_{n}, called the Smith normal form of A. The sequence p^{α_{1}}, …, p^{α_{n}} defines the invariant factors of A, and α = (α_{1}, …, α_{n}) the invariant partition or the sequence of Smith invariants of A. It is known that partitions α, β, γ are respectively the Smith invariants of n by n non-singular matrices A, B, and C such that AB = C if and only if there exists a Littlewood–Richardson tableau T of type (α, β, γ), that is, a tableau of shape γ/α which rectifies to the key-tableau of weight β (or Yamanouchi tableau of weight β) [10, 11]. In other words,

(1.1)  there exists U ∼ I_{n}, Δ_{α}UΔ_{β} ∼ Δ_{γ} if and only if c_{α,β}^{γ} ≠ 0,

where I_{n} denotes the n by n identity matrix, and c_{α,β}^{γ} is the Littlewood–Richardson coefficient of type (α, β, γ). The relationship between the Smith invariants of a product of matrices and the product of Schur functions was noticed earlier, with different approaches, by several authors, see P. Hall, J. A. Green, T. Klein, R. C. Thompson, et al. [1, 2, 3, 14, 16, 29]. (For an overview and other interrelations, see the survey by W. Fulton [11] as well as [10, 12, 13].)

One can show (1.1) by introducing the notion of a matrix realization of a pair (T, K), where T is a skew-tableau which rectifies to the key-tableau K [3, 4, 5]. (We warn the reader that French notation is used.) Such a matrix realization consists of a sequence of products of matrices

(1.2)  Δ_{α}, Δ_{α}UΔ_{(1^{m_{1}})}, Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{2}})}, …, Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{2}})} ⋯ Δ_{(1^{m_{t}})},

where U is an unimodular matrix, α is a partition of length ≤ n, and m = (m_{1}, …, m_{t}) is a non-negative integral vector with m_{i} ≤ n for all i. In particular, (1^{m_{i}}) denotes the partition whose Young diagram is a column of length m_{i}. Transposing the nested sequence of Smith invariants α^{0} = α ⊆ α^{1} ⊆ … ⊆ α^{t}, defined by that sequence of matrices, one gets a tableau T with skew-shape the transpose of α^{t}/α, and weight m. It is shown in [3, 6] that the column-reading word of T, from left to right, is congruent to a key-tableau, and the sequence of column words J_{t} ⋯ J_{2} J_{1}, called the indexing-set word of T, with J_{k} the column-word of length m_{k} comprising the column-indices of the letter k in T, left to right, for k ≥ 1, is a frank word.

The action of the simple transposition s_{i} = (i, i + 1) on (m_{1}, …, m_{t}), by exchanging m_{i} and m_{i+1}, affords an action of s_{i} on the sequence of products of matrices (1.2), by exchanging Δ_{(1^{m_{i}})} and Δ_{(1^{m_{i+1}})}. Therefore the operators {s_{i}} generate an action of the symmetric group on the set of the semistandard Young tableaux defined by those sequences of products of matrices. This action is indeed two-fold: on the one hand, on the tableau column-reading words, one obtains an action of the symmetric group on words congruent to key-tableaux defined by reflection crystal operators based on nonstandard matching of parenthesis; and on the other hand, on the tableau indexing-set words, it yields an action of the symmetric group on frank words. Checking the braid relations for t = 3, these actions turn out to be the ones generated by triples of sets of the ordered set P [n].

The paper is organised as follows. In the next section we analyse the relation between reflection crystal operators on words over a two-letter alphabet and operators defined by jeu de taquin slides on two-column skew-tableaux. An important tool in this analysis is the dual RSK correspondence and its symmetry as well as its interpretation in terms of skew-tableaux. As an application of this duality, we study, in Subsection 2.3, the sequence of Smith invariants, equivalently the skew-tableaux, associated with the sequences (1.2) for t = 2. For this, we have to define two-column frank word variants of jeu de taquin and to show their relationship with non-standard pairing of parentheses on words congruent to two-letter key-tableaux. Moreover, variants of jeu de taquin and non-standard reflection crystal operators have a full interpretation in this matrix context, given in Theorem 2.4.

In Section 3, considering the sequence (1.2) for t = 3, we study the hexagon of skew-tableaux defined by

Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{2}})}Δ_{(1^{m_{3}})} Δ_{α}UΔ_{(1^{m_{2}})}Δ_{(1^{m_{1}})}Δ_{(1^{m_{3}})}  Δ_{α}UΔ_{(1^{m_{2}})}Δ_{(1^{m_{3}})}Δ_{(1^{m_{1}})} Δ_{α}UΔ_{(1^{m_{3}})}Δ_{(1^{m_{2}})}Δ_{(1^{m_{1}})} Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{3}})}Δ_{(1^{m_{2}})}  Δ_{α}UΔ_{(1^{m_{3}})}Δ_{(1^{m_{1}})}Δ_{(1^{m_{2}})}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{2}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{2}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{1}  .

We have two hexagons, one defined by the words of the skew-tableaux, and the other one defined by the indexing-set frank words. Since these hexagons are generated by Smith invariants, the matrix setting imposes conditions on their vertices, given in Lemma 3.2. Theorems 3.6 and 3.8 give a combinatorial interpretation of these conditions and characterize the variant jeu de taquin operators and the non-standard reflection crystal operators satisfying the braid relations, that is, closing these hexagons. Algorithm 3.11 shows that Lascoux–Schützenberger actions of the symmetric group on frank words and words congruent to key-tableaux are respectively obtained: from a particular shuffle decomposition of a three-column tableau; and from a shuffle decomposition of a three-letter Yamanouchi word. Again these actions of the symmetric group have a full interpretation in our matrix setting, given in Theorem 3.14. Finally, in Example 3.15, we exhibit two dual permutahedra in 𝔖_{4}, generated by variants of jeu de taquin operators and reflection crystal operators type based on non-standard pairing.

2. Variants on two-column jeu de taquin, non-standard pairing, and Smith invariants

2.1. Tableaux and dual RSK. A composition m = (m_{1}, …, m_{t}, …) is a (finite or infinite) sequence of non-negative integers, almost all zero. A partition is a weakly decreasing composition, and we denote by β(m) the partition obtained by rearranging the entries of the composition m. It is convenient to not distinguish between two compositions which only differ by a string of zeros at the end. The Young diagram of a partition γ = (γ_{1} ≥ γ_{2} ≥ ⋯ ≥ γ_{n} ≥ 0) is the set of ordered pairs of integers Y (γ) = {(i, j) : 1 ≤ i ≤ γ_{j}, 1 ≤ j ≤ n}. We identify a partition with its Young diagram. The conjugate partition γ′ = (γ′_{1}, γ′_{2}, …) of γ is the partition corresponding to the transpose of Y (γ). A skew-diagram γ/α is the set difference Y (γ) − Y (α) of Young diagrams of partitions. A (semistandard) (skew) tableau T over the alphabet [t] = {1, …, t} is a function (often called filling) T : Y ⟶ [t] from a (skew) diagram Y to the positive integers in [t] such that they are weakly increasing in each row and strictly increasing up each column. The shape shape(T) of T is the domain of the tableau T. The columnreading word of the (skew) tableau T is w = w^{1}w^{2} … where w^{i} is the column-word (strictly decreasing word) over [t] comprising the ith column of T, left to right. Define the column-shape of T to be the composition colshape(T) = (|w^{1}|, |w^{2}|, …), where |w| denotes the length of w. Similarly, the row-reading word of T is ⋯ u^{2}u^{1}, where u^{i} is the row word (increasing word) over [t] comprising the ith row of T, bottom to top, and the row-shape of T is the composition (|u^{1}|, |u^{2}|, …). A vertical strip is a skew-tableau with rows of length at most one. The weight of T is the weight of the word w, that is, the sequence (m_{1}, …, m_{t}) where m_{i} is the number of occurrences of the letter i in w. The column-indexing-set word of T is J = J_{t} ⋯ J_{2}J_{1} where J_{k} is the column-word of length m_{k} defined by the set of column-indices of the letter k in T, left to right. An example of a Young tableau of skew-shape (4, 3, 2, 2)/(3, 1) is

(2.1)  T = \begin{matrix}4 & 4 \\ 2 & 2 \\ • & 1 & 3 \\ • & • & • & 2 \end{matrix}.

The column-reading word is 42 421 3 2, the column-indexing-set word is J = 21 3 421 2 and the weight (1, 3, 1, 2).

A standard tableau is a tableau filled with the numbers {1, …, n} where n is the number of cells or boxes in the (skew) Young diagram. The standardization std(T) of a Young tableau T is the standard tableau obtained by simultaneously replacing the 1’s in T from left to right by 1, 2, …, m_{1}, the 2′s by m_{1} + 1, …, m_{1} + m_{2} etc., where (m_{1}, …, m_{t}) is the weight of T. The transpose of a standard tableau T is still a standard tableau, of conjugate shape, and we denote it by T^{τ}.

Let K(m) denote the key-tableau of weight m, that is, the tableau of weight m whose column-shape is β′(m). There is an obvious bijection between compositions and keytableaux since K(m) is the tableau whose first m_{j} columns contain the letter j, for all j [26].

The overlap of a pair (u, v) of column-words u and v is the maximum number of rows of length two obtained with respect to shifting the column u down with respect to the column v. For instance, the overlap of the first two columns of the skew-tableau (2.1) is two, of the second and the third is one, and of the last two is zero. A skew-tableau is said to be in compact form if the number of rows of size two of any two consecutive columns is the overlap of the words comprising those columns. Thus a skew-tableau T in compact form is identified with its column-reading word seen as a sequence of column words. For instance the skew-tableau (2.1) is in compact form.

Let [t]^{∗} denote the free monoid over the alphabet [t] = {1, …, t}. The Knuth or plactic congruence ≡ [19, 9, 17, 23] on the words over the alphabet [t] is the congruence in [t]^{∗} defined by the transitive closure of the relations below, where x, y and z are letters in [t], and a and b are words in [t]^{∗}:

axzyb ≡ azxyb, x ≤ y < z,
ayzxb ≡ ayxzb, x < y ≤ z.

Given the word w = x_{1} ⋯ x_{N}, the row insertion of w produces a unique pair of tableaux (P (w), Q(w)) of the same shape. P (w) is the unique tableau of partition shape whose column-reading word is Knuth equivalent to w, and Q(w), often called the Q-symbol of w, is the standard tableau of the same shape as P (w) such that shape(Q(w)|_{[i]}) = shape(P (x_{1} ⋯ x_{i})), 1 ≤ i ≤ t, where Q(w)|_{[i]} denotes the restriction of the tableau Q(w) to the letters in [i]. Two (skew) tableaux are said to be Knuth equivalent if their words are congruent, equivalently, one is obtained from the other by jeu de taquin slides [9, 23, 28].

Let v = v^{1}v^{2} … v^{n}, with each v^{i} a column-word (some of the v^{i} may be the empty word), be an n-column factorization of v. The factorization v^{1}v^{2} … v^{n} may be identified, as a sequence of column words, with the skew-tableau in compact form where v^{i} is the word comprising the i-th column. The column-shape of v^{1}v^{2} … v^{n}, denoted by colshape(v^{1}v^{2} … v^{n}), is the column-shape of that skew-tableau. A factorization is minimal if the number of factors is minimal. We write colshape(v) (unique up to zeros) to mean the column-shape of the minimal factorization of v. Lascoux and Schützenberger called v frank word [21] if it has a factorization of column-shape (in fact minimal) a permutation of the non-zero parts of the conjugate shape of P (v).

The dual RSK correspondence [17, 9] can be defined as a bijection from the set of finite sequences of column-words (⋯ J_{2}J_{1}) to pairs of tableaux (Q, P) of conjugate shapes, where Q = P (⋯ J_{2}J_{1}) and shape(P |_{[i]}) = shape(P (J_{i} ⋯ J_{1}))′ for all i.

Let A and B be two totally ordered alphabets. We consider biwords of pairwise distinct biletters

(\begin{matrix}u_{1} & ⋯ & u_{k} \\ v_{1} & ⋯ & v_{k} \end{matrix}) ,

with u_{i} ∈ A and v_{j} ∈ B. The biword is said to be in anti-lexicographic order if its biletters \binom{u_{i}}{v_{i}} satisfy u_{i} < u_{i+1} or u_{i} = u_{i+1} and v_{i} > v_{i+1}.

Let J = {J_{t−i+1}}_{1≤i≤t}, be a sequence of column-words over the alphabet [n] in dual RSK correspondence with the tableau-pair (Q, P) of conjugate shapes. Let m_{i} be the length of J_{i}, 1 ≤ i ≤ t, and consider the biword with no two identical biletters,

(2.2)  Σ′ = (\begin{matrix}J_{t} & ⋯ & J_{1} \\ t^{m_{t}} & ⋯ & 1^{m_{1}} \end{matrix}) .

By increasing rearrangement of Σ′ with respect to the anti-lexicographic order, we get

(2.3)  Σ = (\begin{matrix}1^{f_{1}} & ⋯ & n^{f_{n}} \\ w^{1} & ⋯ & w^{n} \end{matrix}) ,

with w^{i} a column word of length f_{i}. Clearly the word J_{t} ⋯ J_{1} can be seen as the columnindexing-set word of a skew-tableau with column reading word w^{1} ⋯ w^{n}, and conversely w^{n} ⋯ w^{1} as the column-indexing-set word of a skew-tableau with column-reading word J_{1} ⋯ J_{t}. Therefore, the dual RSK correspondence defines also a bijection between skewtableaux in compact form and tableau-pairs of conjugate shapes. For instance, consider the skew-tableau (2.1). Construct Σ and Σ′ accordingly. The bottom word of Σ is w and the top word of Σ′ is J,

Σ = (\begin{matrix}11 22 3 44 \\ 42 21 3 32 \end{matrix}) ⟷ Σ′ = (\begin{matrix}1 43 421 2 \\ 4 33 222 1 \end{matrix}) .

From Proposition 5, the symmetry theorem in [9, Appendix A.4.3], we have the following result.

Theorem 2.1. Given the sequence J = {J_{t−i+1}}_{1≤i≤t}, of column-words over the alphabet [n], there is one and only one word w over the alphabet [t], with weight the reverse of the column shape of {J_{t−i+1}}_{1≤i≤t}, revcolshape({J_{t−i+1}}_{1≤i≤t}), such that

(a) P (w) = P.

(b) the Q-symbol or recording tableau of w satisfies Q(w) = std(Q)^{τ} with Q = P (J). Moreover J is a frank word of colshape({J_{t−i+1}}_{1≤i≤t}) if and only if P =

K(revcolshape(w)) [21, 27].

As frank words, in a congruence class, are completely determined by their Q-symbols, it follows that frank words, in a plactic class, are in bijection with the set of permutations of the non-zero parts of the conjugate shape of the tableau in that class [21] and we have an action of the symmetric group on frank words. Lascoux and Schützenberger have translated this action of the symmetric group on frank words in the language of jeu de taquin slides on two-column skew-tableaux either aligned in the bottom or in the top. This jeu de taquin operation can be extended to any two-column skew-tableaux in compact form to define an action of the symmetric group on skew-tableaux (see [22]).

2.2. Key-tableaux and sequences of matrix products over a local principal ideal domain. Let the symmetric group 𝔖_{t} act on finite compositions m = (m_{1}, …, m_{t}) via the left action s_{i}m = (m_{1}, …, m_{i+1}, m_{i}, …, m_{t}), with s_{i}, 1 ≤ i ≤ t−1, the simple transpositions of 𝔖_{t}. Then β(m) is the unique partition in the orbit 𝔖_{t}m and β′(m) its conjugate. Recall that the column-shape of the key-tableau K(m) is β′(m). Considering the bijection between compositions and key-tableaux mentioned earlier, we identify K(m) with the sequence of diagonal matrices (Δ_{(1^{m_{1}})}, …, Δ_{(1^{m_{t}})}) in the sense that the nested sequence of partitions (1^{m_{1}}) ⊆ (1^{m_{1}}) + (1^{m_{2}}) ⊆ … ⊆ (1^{m_{1}}) + … + (1^{m_{t}}) = β′(m) defines the key-tableau K(m) and, simultaneously, are the Smith invariants of the sequence of products of matrices Δ_{(1^{m_{1}})}, Δ_{(1^{m_{1}})}Δ_{(1^{m_{2}})}, …, Δ_{(1^{m_{1}})}Δ_{(1^{m_{2}})} ⋯ Δ_{(1^{m_{t}})} = Δ_{β′(m)}. For instance,

K(10325) = 5 4 5 3 4 5 is identified with (Δ_{[1]}, Δ_{∅}, Δ_{[3]}, Δ_{[2]}, Δ_{[5]}). 1 3 3 5 5

Let U be an n by n unimodular matrix, that is, U ∼ I_{n}. Put Δ_{α}UK(m) for the sequence

Δ_{α}, Δ_{α}UΔ_{(1^{m_{1}})}, Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{2}})}, …, Δ_{α}UΔ_{(1^{m_{1}})}Δ_{(1^{m_{2}})} ⋯ Δ_{(1^{m_{t}})} = Δ_{α}UΔ_{β′(m)}.

The nested sequence of Smith invariants α^{0} = α ⊆ α^{1} ⊆ … ⊆ α^{t} = γ defined by this sequence of products of matrices is such that α^{i+1}/α^{i} is a vertical strip of length m_{i+1}, for i = 0, 1, …, t − 1. Thus Δ_{α}UK(m) is identified with the tableau T of shape γ′/α′ and weight m, with indexing-set words J_{k+1} = {i : α_{i}^{k+1} = α_{i}^{k} + 1}, 1 ≤ k ≤ t − 1. It is shown in [6] that the column-reading word w of T satisfies P (w) = K(m) and thus, from Theorem 2.1, J is a frank word of shape the non-null parts of the reverse of m.

When we consider the action of the symmetric group on finite compositions m we are simultaneously defining an action of the symmetric group on sequences of matrices Δ_{α}UK(m), where U is a fixed unimodular matrix and α a fixed partition, and therefore on tableaux of skew-shape whose rectifications are the key-tableaux K(m). Thus, we obtain two families of actions of the symmetric group which are translations of each other: one over non-congruent frank words running over tableaux of column-shape β(m) where the weight is a permutation of the entries of m; and the other one on words congruent to key-tableaux K(m), not necessarily sharing the same Q-symbol. This duality is explained in the next section.

2.3. Jeu de taquin on two-column words and reflection crystal operators. It is known that there is a duality between reflection crystal operators and jeu de taquin on two-column skew tableaux [18, 22]. The jeu de taquin slides exchanging the length of two consecutive columns i, i + 1 of a t-column skew-tableau in compact form, counting right to left, is translated, by the dual RSK-correspondence, into the i-th reflection crystal operator on words over the alphabet [t]. In the particular case of frank words it is translated into the i-th reflection crystal operator on words congruent to key-tableaux. Define the operator Θ on a two-column skew-tableau T = J_{2}J_{1} in compact form and row-shape (1^{s}, 2^{q}, 1^{r}), for some q, r, s ≥ 0, as follows. If r > s (r < s), perform jeu de taquin slides in the first |r − s| inside (outside) corners marked ■ of the skew-tableau T until they become outside (inside) corners in the second (first) column. In other words, we slide down (up) maximally up to |r − s| positions the entries of the first (second) column; then we exchange the east (west) entries with the vacant neighbours ■. Then Θ T = T′ is a two-column skew-tableau in compact form with row-shape (1^{r}, 2^{q}, 1^{s}). Obviously Θ T′ = T. In particular, when r = 0 or s = 0, Θ is the jeu de taquin on frank words. For instance, the jeu de taquin slides, with respect to the corner ■ as below, define the operator Θ on T of row shape (1, 2^{3}, 1^{2}), and on T′ of row shape (1^{2}, 2^{3}, 1)

(2.4)  T = \begin{aligned}7 & & & & 7 \\ 3 & 6 & & & 6 & ■ \\ 2 & 5 & Θ & & 3 & 5 \\ 1 & 4 & ⟷ & T′ = & 2 & 4 \\ ■ & 3 & & & 1 & 3 \\ & 2 & & & & 2 \end{aligned} .

Let T be a t-column skew-tableau in compact form. Define the operator Θ_{i} on T as follows: apply Θ to the columns i and i + 1 of T, counting right to left, and put the outcome t-column skew-tableau in compact form. As jeu de taquin preserves Knuth equivalence, we have Θ_{i}T ≡ T.

Let w = w_{1}w_{2} … w_{g}, w_{i} ∈ [t], be a word. An r-pairing of w is a set of indexed pairs (called r-pairs) (w_{i}, w_{j}) such that 1 ≤ i < j ≤ k, w_{i} = r + 1, and w_{j} = r, and if (w_{l}, w_{s}) is another pair, then i, l, j, s are pairwise distinct. View each r + 1 (respectively r) as a left (respectively right) parenthesis and ignore the other letters. The r-pairs of w are precisely the matched parentheses. Furthermore the subword of unpaired r′s and (r + 1)′s must be a subword of w of the form r^{k}(r + 1)^{l}. In general, not every r-pairing gives the maximal number of r-pairs of w, and if θ̃_{r} is the operator which replaces the word r^{k}(r + 1)^{l} of unpaired r′s and (r + 1)′s in w (in the corresponding positions) by r^{l}(r + 1)^{k}, unless certain conditions are imposed on the r-pairing, the maximal number of r-pairs of θ̃_{r}w and w may be different. For example, w = 12112 has two 1-pairings with a maximal number of 1-pairs, and θ̃_{1}w = 1(21)22 has just one 1-pairing. However, when either k = 0 or l = 0 they have always the same maximal number of r-pairs. In this case, as we shall see in the next subsection, the operator θ̃_{r} can be reduced to a variant of jeu de taquin on two-column frank words.

The standard r-pairing on w is the particular r-pairing obtained in the following way. Start with the subword w′ = x_{1} ⋯ x_{m}, the restriction of w to the alphabet {r, r + 1}. Then, bracket every factor r + 1 r of w′. The letters which are not bracket constitute a subword w′′ of w′. Then, bracket every factor r + 1 r of w′′. Continue this procedure until it stops. It remains a word of the form r^{k}(r + 1)^{l}. The reflection crystal operator, based on the standard r-pairing, denoted by θ_{r}, which replaces the word r^{k}(r + 1)^{l} of unpaired r′s and (r + 1)′s in w by r^{l}(r + 1)^{k}, can be reduced to jeu de taquin slides on two-column skew tableaux.

Recalling the definition of the biwords (2.2) and (2.3), the operators θ_{i} on words over the alphabet [t], and Θ_{i} on t-column skew-tableaux J = {J_{t−i+1}}_{1≤i≤t}, in compact form, are a translation of each other in the sense of the following commutative diagram

(2.5)  Σ = ( \begin{matrix}J ↑ \\ w \end{matrix} )  Σ′ = ( \begin{matrix}⋯ J_{i+1} J_{i} ⋯ \\ ⋯ (i + 1)^{q+k} i^{q+l} ⋯ \end{matrix} ) Σ̃ = ( \begin{matrix}J ↑ \\ θ_{i}w \end{matrix} )  Σ̃′ = ( \begin{matrix}⋯ Θ_{i}(J_{i+1} J_{i}) ⋯ \\ ⋯ (i + 1)^{q+l} i^{q+k} ⋯ \end{matrix} )^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ,

where J ↑ indicates J by weakly increasing order.

As column words may be identified with their support sets, these operators can be formulated in P [n] the set of all subsets of [n]. We consider on P [n] two orders, one by letting B ≤ B′ whenever there is an increasing injection i : B → B′, that is, x ≤ i(x), and the other one by putting B ▷ B′ whenever there exists a decreasing injection B ← B′ : j, that is, j(x) ≤ x.

Define the skew-tableau in compact form \begin{aligned}A \\ B & C \\ & D \end{aligned} , where A ∪ B, C ∪ D are columns

such that B ≤ C, |B| = |C| = q, |A| = s, and |D| = r. Suppose r > s. Slide down maximally the column B along C ∪ D such that the weakly increasing order in rows is preserved, and define the column X ⊆ C ∪ D whose entries have west adjacent neighbours in B. Then θ_{i} can be based in any pairing of parentheses defined by any increasing injection j : B → X. We have the equivalence, Θ_{i}T ≡ T if and only if the operator θ_{i} preserves the Q-symbol, that is, Q(w) = Q(θ_{i}w). For instance, for the two-column tableau T (2.4), we have the following diagram

Σ = ( \begin{matrix}1 2 2 3 3 4 5 6 7 \\ 2 2 1 2 1 1 1 1 2 \end{matrix} )  Σ′ = ( \begin{matrix}T \\ 2^{3+1} 1^{3+2} \end{matrix} ) Σ̃ = ( \begin{matrix}1 2 2 3 3 4 5 6 7 \\ 2 2 1 2 1 1 1 2 2 \end{matrix} )  Σ̃′ = ( \begin{matrix}Θ_{1}T \\ 2^{3+2}1^{3+1} \end{matrix} )^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ,

where w = (2(21)(21)1)1^{2}2 → θ_{1}w = (2(21)(21)1)12^{2}.

From the diagram (2.5), we have the following result.

Theorem 2.2. The following statements are equivalent:

(a) The operators Θ_{i}, 1 ≤ i ≤ t − 1, define an action of the symmetric group 𝔖_{t} on the set of t-column words, equivalently, on the t-column skew-tableaux in the compact form. Moreover, Θ_{i}T ≡ T, 1 ≤ i ≤ t − 1.

(b) [19, 23]The operators θ_{i}, 1 ≤ i ≤ t − 1, define an action of the symmetric group on all words over the alphabet [t], and preserve the Q-symbol.

2.4. Two-column frank word variants of jeu de taquin, pairing of parentheses, and Smith invariants. In this section, we define variants of jeu de taquin on twocolumn frank words and show its relationship with a non-standard pairing of parentheses on words congruent to two-letter key-tableaux. As an application we describe the Smith invariants, equivalently, the skew-tableaux on a two-letter alphabet, associated with the sequences Δ_{α}UK(m) and Δ_{α}UK(s_{1}m) with m = (m_{1}, m_{2}) and s_{1} the elementary transposition (1 2).

Restrict the jeu de taquin operation Θ to a two-column tableau or contre-tableau (a two-column skew-tableau such that the pair of columns is aligned at the top) J_{2} J_{1} [9, 28], and denote by Θ̃ a variant of Θ which runs as follows. If J_{2}J_{1} is a contre-tableau (tableau), slide vertically the entries of the column J_{2} (J_{1}) along the column J_{1} (J_{2}) such that the row weakly increasing order is preserved, and every common label to the two columns never has a vacant west (east) neighbour. Then exchange the vacant positions with the east (west) neighbours. In particular, when the first (second) column J_{2} (J_{1}) is slid down (up) maximally such that the row weakly increasing order is preserved, we get the outcome of the jeu de taquin operation. For instance,

(2.6)  Θ : \begin{matrix}2 & 5 \\ 1 & 4 \\ ■ & 3 \\ ■ & 2 \end{matrix} ⟷ \begin{matrix}■ & 5 \\ ■ & 4 \\ 2 & 3 \\ 1 & 2 \end{matrix} ⟷ \begin{matrix}5 & ■ \\ 4 & ■ \\ 2 & 3 \\ 1 & 2 \end{matrix} ,

(2.7)  Θ̃ : \begin{matrix}2 & 5 \\ 1 & 4 \\ ■ & 3 \\ ■ & 2 \end{matrix} ⟷ \begin{matrix}■ & 5 \\ 2 & 4 \\ ■ & 3 \\ 1 & 2 \end{matrix} ⟷ \begin{matrix}5 & ■ \\ 2 & 4 \\ 3 & ■ \\ 1 & 2 \end{matrix} ⟷ \begin{matrix}5 & ■ \\ 3 & ■ \\ 2 & 4 \\ 1 & 2 \end{matrix} .

Obviously, Θ̃(J_{2}J_{1}) and Θ(J_{2}J_{1}) are not congruent unless Θ̃ = Θ, but Θ̃(J_{2}J_{1}) is a frank word with the same shape and weight as Θ(J_{2}J_{1}).

Suppose that w is congruent to the key-tableau of weight (0^{r−1}, m_{r}, m_{r+1}). Without loss of generality, assume m_{r+1} ≤ m_{r}. Let J_{r+1}J_{r} be a frank word of shape

(m_{r+1}, m_{r}, 0^{r−1}), such that sorting the biletters of the biword Σ′ = ( \begin{matrix}J_{r+1}J_{r} \\ (r + 1)^{m_{r+1}}r^{m_{r}} \end{matrix} ),

by weakly increasing rearrangement of the biletters for the anti-lexicographic order, we

get Σ = ( \begin{matrix}J_{r+1}J_{r} ↑ \\ w \end{matrix} ) . Consider an r-pairing in w defined by an increasing injection

i : J_{r+1} ⟶ J_{r} such that J_{r} ∩ J_{r+1} ⊆ i(J_{r+1}). To perform θ̃_{r}w based on this r-pairing means to apply an operator Θ̃ on J_{r+1}J_{r} (denoted by Θ̃_{r}) which exchanges the vacant entries of the first column with the corresponding east neighbours consisting of J_{r} \ i(J_{r+1}) in the second column J_{r}. Conversely, an operator Θ̃_{r} on J_{r+1}J_{r} means an operator θ̃_{r} on w, where the r-pairing on w is defined by any increasing injection i : J_{r+1} ⟶ J_{r} such that Θ̃J_{r+1}J_{r} = [J_{r+1} ∪ (J_{r} \ B)]B, where J_{r} ∩ J_{r+1} ⊆ i(J_{r+1}) = B. When Θ̃_{r} = Θ_{r} we get the standard pairing of parentheses on w and hence θ_{r}. Thus the operators Θ̃_{r}, Θ_{r} and θ̃_{r}, θ_{r} are respectively translated into each other, according the following commutative diagram,

(2.8)  Σ = ( \begin{matrix}J_{r+1}J_{r} ↑ \\ w \end{matrix} )  Σ′ = ( \begin{matrix}J_{r+1}J_{r} \\ (r + 1)^{m_{r+1}}r^{m_{r}} \end{matrix} ) Σ̃ = ( \begin{matrix}J_{r+1}J_{r} ↑ \\ θ̃_{r}w \end{matrix} )  Σ̃′ = ( \begin{matrix}Θ̃(J_{r+1}J_{r}) \\ (r + 1)^{m_{r}}r^{m_{r+1}} \end{matrix} )^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  .

If the row insertion of w gives the pair of tableaux (P, Q) then the row insertion of θ̃_{r}w leads to the pair (θ_{r}P, Q′), where Q and Q′ are distinct tableaux unless θ̃_{r} = θ_{r}. As a Knuth class is not closed under the action of the operator Θ̃_{r}, θ̃_{r} does not preserve the Q-symbol but we have θ_{r}w ≡ θ̃_{r}w. For instance, in (2.6), any increasing injection {1, 2} → {2, 3} defines a standard pairing of parentheses, giving rise to θ_{1} : (2(21)1)1 → (2(21)1)2; and in (2.7), any increasing injection {1, 2} → {2, 4} defines a pairing of parentheses, giving rise to θ̃_{1} : (2(21)11) → (2(21)21).

We are now in the position to describe the Smith invariants, equivalently, the skewtableaux on a two-letter alphabet associated with the sequences Δ_{α}UK(m) and Δ_{α}UK(s_{1}m). We let P_{σ} be the permutation matrix associated with the permutation σ ∈ 𝔖_{n}.

Lemma 2.3. [5] (a) Let U be an n by n unimodular matrix. Then, there exists σ ∈ S_{n} such that U = T P_{σ}QL, where T is an n by n upper triangular matrix, with 1′s along the main diagonal, Q is an n by n upper triangular matrix, with 1′s along the main diagonal, and multiples of p above it, and L is an n by n lower triangular matrix, with units along the main diagonal.

(b) By elementary operations on the left and on the right, Δ_{α}UK(m) may be considered equal to Δ_{α}P_{σ}QK(m), with σ ∈ 𝔖_{n}.

(c) The Smith normal form of Δ_{α}P_{σ}QD_{[m_{1}]}, with σ ∈ 𝔖_{n}, is the diagonal matrix Δ_{α^{1}} where α ⊆ α^{1} is a vertical strip of length m_{1}.

Theorem 2.4. [5] Let T and T′ be respectively the tableaux defined by the sequences Δ_{α}UK(m) and Δ_{α}UK(s_{1}m), with indexing-set words J_{2}J_{1}, J′_{2}J′_{1} respectively, and words w, w′ respectively. Then,

(a) J_{2}J_{1}, J′_{2}J′_{1} are frank words such that Θ̃_{1}J_{2}J_{1} = J′_{2}J′_{1}, for some operator Θ̃_{1}.

(b) w ≡ K(m) and w′ = θ̃_{1}w ≡ K(s_{1}m), for some operator θ̃_{1}.

Conversely, if T and T′ are respectively tableaux with indexing-set frank words J_{2}J_{1} and J′_{2}J′_{1} satisfying J′_{2}J′_{1} = Θ̃_{1}J_{2}J_{1}, then there exists an unimodular matrix U such that Δ_{α}UK(m) and Δ_{α}U′K(s_{1}m) define the tableaux T and T′ respectively.

Example 2.5. Let U = P_{4321}T_{14}(p), where T_{14}(p) is the elementary matrix obtained from the identity by placing the prime p in position (1, 4). With α = (2, 1) the sequences

Δ_{α}UK(3, 2) and Δ_{α}UK(2, 3) define, respectively, T = \begin{matrix}2 \\ • & 1 & 2 \\ • & • & 1 & 1 \end{matrix} and T′ = \begin{matrix}2 \\ • & 2 & 2 \\ • & • & 1 & 1 \end{matrix}.

The words w = 21211 of T and w′ = 22211 of T′ satisfy θ̃_{1}w = w′ ≡ θ_{1}w, where θ̃_{1} is the operator based on the parentheses matching (21(21)1). However, if we choose U′ = P_{3241}T_{24}(p), the sequences Δ_{α}U′ K(3, 2) and Δ_{α}U′ K(2, 3) define, respectively,

T and T′′ = \begin{matrix}2 \\ • & 1 & 2 \\ • & • & 1 & 2 \end{matrix}. In this case, the word w′′ of T′′ satisfy θ_{1}w = w′′. The

corresponding operations on the indexing frank words are given as follows:

(2.9)  Θ : \begin{matrix}■ & 4 \\ 3 & 3 \\ 1 & 2 \end{matrix} ⟷ \begin{matrix}4 & ■ \\ 3 & 3 \\ 1 & 2 \end{matrix}  Θ̃ : \begin{matrix}3 & 4 \\ 1 & 3 \\ ■ & 2 \end{matrix} ⟷ \begin{matrix}3 & 4 \\ 1 & 3 \\ 2 & ■ \end{matrix} ⟷ \begin{matrix}3 & ■ \\ 2 & 4 \\ 1 & 3 \end{matrix} .

The operators Θ_{r} (θ_{r}) can be extended to frank words with more than two columns (words on a t-letter alphabet, t ≥ 2) [19, 23]. Under certain conditions, operators Θ̃_{r} (θ̃_{r}) can be extended, as well, to frank words with more than two columns (words on a t-letter alphabet, t ≥ 2). For this, we generalize a criterion, by Lascoux and Schützenberger in [21], to test whether the concatenation of a frank word with a column word is a frank word. Denote, respectively, by L(J) and R(J) the left and right columns of a frank word J.

Theorem 2.6. [21] The concatenation JJ′ of two frank words J and J′ is frank if and only if R(H)L(H′) is frank for any pair of frank words H, H′ such that H ≡ J and H′ ≡ J′.

Notice that when J, J′ are column-words, JJ′ is frank if and only if JJ′ is a tableau or a contre-tableau. Therefore, we deduce the following criterion for the concatenation of a column with a frank word.

Corollary 2.1. Let J = J_{k} ⋯ J_{1} be a frank word and J_{k+1} a column. Then, J_{k+1}J is frank if and only if J_{k+1}J_{k} and \overline{J_{k}}J_{k−1} ⋯ J_{1} are frank words, where \overline{J}_{k+1} \overline{J}_{k} = Θ_{k}(J_{k+1}J_{k}).

The criterion given by this corollary can be generalized to operators Θ̃.

Corollary 2.2. Let J = J_{k} ⋯ J_{1} be a frank word and J_{k+1} a column. Then, J_{k+1}J is frank if and only if J_{k+1}J_{k} and J̃_{k}J_{k−1} ⋯ J_{1} are frank words, where J̃_{k+1} J̃_{k} = Θ̃_{k}(J_{k+1}J_{k}) for some operator Θ̃_{k}.

Proof. The necessary condition is a consequence of the previous corollary. Conversely, assume the existence of an operator Θ̃_{k} in the required conditions, and let \overline{J}_{k+1} \overline{J}_{k} = Θ_{k}(J_{k+1}J_{k}). Clearly, we have \overline{J}_{k} ≤ J̃_{k}, and also \overline{J}_{k+1} ▷ J̃_{k+1}, since |\overline{J}_{k}| = |J̃_{k}|. By the hypotheses, the product J̃_{k}L(H) is frank, for any frank word H ≡ J_{k−1} ⋯ J_{1}. This means that either J̃_{k} ≤ L(H), or J̃_{k} ▷ L(H). By transitivity, we find that either \overline{J}_{k} ≤ L(H), or \overline{J}_{k} ▷ L(H), i.e., \overline{J}_{k}L(H) is frank. Thus, by Theorem 2.6, the word \overline{J}_{k}J_{k−1} ⋯ J_{1} is frank, and therefore, by the previous corollary, J_{k+1}J is frank. □

The following theorem was proved in [6]. Here we give a different proof based on indexing-set words.

Theorem 2.7. Let T be the tableau defined by Δ_{α}U K(m), with word w and J the indexing set word. Then P (w) = K(m) and J is a frank word of column-shape the reverse of m.

Proof. Let J = J_{t} … J_{1} with column-shape the reverse of m. We will prove, by induction on t ≥ 1, that J_{t} ⋯ J_{1} is a frank word. When t = 1 the result is trivial, and the case t = 2 is a consequence of Theorem 2.4 (see [5]). So, let t > 2 and let T be the tableau defined by Δ_{α}UK(m_{1}, …, m_{t}). By the inductive step, the word J_{t−1} ⋯ J_{1} is frank, since the sequence Δ_{α}UK(m_{1}, …, m_{t−1}) defines the tableau T′ with indexing-set word J_{t−1} … J_{1} and weight (m_{1}, …, m_{t−1}).

By the Smith normal form theorem (see for instance [25]), there is a partition ᾱ and an unimodular matrix U′ such that by elementary row operations, ΔᾱUD_{[m_{1}]} ⋯ D_{[m_{t−2}]} can be reduced to ΔᾱU′. The sequence ΔᾱU′K(m_{t−1}, m_{t}) defines the tableau \overline{T} with indexing-set word J_{t−1}, J_{t}, and weight (m_{t−1}, m_{t}). By the case t = 2, the word J_{t}J_{t−1} is frank. Moreover, by Theorem 2.4, we find that if \overline{T}′ is the tableau defined by the sequence ΔᾱU, K(m_{t}, m_{t−1}), the indexing sets \overline{J}_{t−1}, \overline{J}_{t} of \overline{T}′ satisfy \overline{J}_{t}\overline{J}_{t−1} = Θ̃_{t−1}(J_{t}J_{t−1}) for some operator Θ̃_{t−1}.

Finally, notice that Δ_{α}UK(m_{1}, …, m_{t−2}, m_{t}) defines the tableau T̃ with indexingset word \overline{J}_{t−1} J_{t−2} … J_{1}, and weight (m_{1}, …, m_{t−2}, m_{t}). By the inductive step, \overline{J}_{t−1}J_{t−2} ⋯ J_{1} is a frank word. Thus, by Corollary 2.2, the word J_{t} ⋯ J_{1} is frank, and therefore, w ≡ K(m). □

3. An action of the symmetric group on Young tableaux of skew-shape

Let s_{i} denote the elementary transposition (i, i + 1) of 𝔖_{t}, 1 ≤ i ≤ t. Let U be an n by n unimodular matrix and β = (β_{1}, β_{2}, β_{3}) a partition. We consider the following hexagon (3.1)

Δ_{α}UK(β_{1}, β_{2}, β_{3}) Δ_{α}UK(β_{2}, β_{1}, β_{3})  Δ_{α}UK(β_{2}, β_{3}, β_{1}) Δ_{α}UK(β_{3}, β_{2}, β_{1}) Δ_{α}UK(β_{1}, β_{3}, β_{2})  Δ_{α}UK(β_{3}, β_{1}, β_{2})^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{2}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} s_{2}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} s_{1}  .

From the discussion in Section 2.2, we may look at (3.1) as a hexagon whose vertices are tableaux of skew-shape such that the words are congruent to a key-tableau K(β_{i_{1}}, β_{i_{2}}, β_{i_{3}}), and the indexing-set frank words have column-shape the reverse of (β_{i_{1}}, β_{i_{2}}, β_{i_{3}}) with (β_{i_{1}}, β_{i_{2}}, β_{i_{3}}) running over the orbit 𝔖_{3}β. Therefore, we have two hexagons, one defined by the words of the skew-tableaux and the other one defined by the indexing-set frank words. These hexagons are copies of each other since operations based on pairing of parentheses can be reduced to variations of the jeu de taquin on two-column frank words and vice versa. Taking into account Theorems 2.4 and 2.7, the next statement follows from the hexagon above. Given σ ∈ 𝔖_{t}, rev denotes the longest permutation of 𝔖_{t}. Let α_{1}, α_{2} be two operators satisfying the Moore–Coxeter relations of the symmetric group 𝔖_{3}, we put ⟨α_{1}, α_{2}⟩ := {α_{i}α_{j} … α_{k} : i j … k ∈ [2]^{∗}}.

Theorem 3.1. Let σ ∈ ⟨s_{1}, s_{2}⟩, θ ∈ ⟨θ_{1}, θ_{2}⟩ and Θ ∈ ⟨Θ_{1}, Θ_{2}⟩ with the same reduced word. Let σT be the tableau defined by Δ_{α}UK(σβ), with word σw and indexing-set frank word σJ of column-shape the reverse of σβ. Then {σT : σ ∈ ⟨s_{1}, s_{2}⟩} are the vertices of a hexagon such that:

(a) there exist operators θ̃_{1} and θ̃_{2} satisfying the Moore–Coxeter relations of the symmetric group 𝔖_{3}, where θ̃ ∈ ⟨θ̃_{1}, θ̃_{2}⟩, with the same reduced word as θ, verifies σw = θ̃w ≡ θK(β) = K(σβ),

(3.2)  w  θ̃_{1}w  θ̃_{2}θ̃_{1}w θ̃_{1}θ̃_{2}θ̃_{1}w θ̃_{2}w  θ̃_{1}θ̃_{2}w^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  .

(b) there exist operators Θ̃_{1} and Θ̃_{2} satisfying the Moore–Coxeter relations of the symmetric group 𝔖_{3}, where Θ̃ ∈ ⟨Θ̃_{1}, Θ̃_{2}⟩, with the same reduced word as Θ, verifies σJ = Θ̃J,

(3.3) J = J_{3}J_{2}J_{1}  Θ̃_{1}J = J_{3}G_{2}G_{1}  Θ̃_{2}Θ̃_{1}J = F_{3}F_{2}G_{1}  Θ̃_{1}Θ̃_{2}Θ̃_{1}J = F_{3}XH_{1} Θ̃_{2}J = L_{3}L_{2}J_{1}  Θ̃_{1}Θ̃_{2}J = L_{3}H_{2}H_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  .

Our aim is to describe explicitly the operators θ̃_{i} and Θ̃_{i} closing respectively the hexagons (3.2) and (3.3), that is, those operators θ̃_{i} and Θ̃_{i} such that θ̃_{1}θ̃_{2}θ̃_{1}w = θ̃_{2}θ̃_{1}θ̃_{2}w and Θ̃_{1}Θ̃_{2}Θ̃_{1}J = Θ̃_{2}Θ̃_{1}Θ̃_{2}J respectively.

In fact the hexagon (3.1) and, hence, hexagon (3.3), obey the following conditions. (The translation of these conditions to hexagon (3.2) will be done later.)

Lemma 3.2. [5] The operators closing hexagons (3.1) and (3.3) obey the following conditions:

(a) If L_{3}L_{2} and F_{3}F_{2} are, respectively, the indexing-set frank words of Δ_{α}UK(β_{1}, β_{3}) and Δ_{α}UK(β_{2}, β_{3}), then there holds the inequality F_{2} ≤ L_{2}.

(b) If L_{3}H_{2} and J_{3}G_{2} are, respectively, the indexing-set frank words of Δ_{α}UK(β_{3}, β_{1}) and Δ_{α}UK(β_{2}, β_{1}), then there holds the inequalityG_{2} ≤ H_{2}.

(c) The operators Θ̃_{1} and Θ̃_{2} defining the hexagon (3.3) are such that Θ̃_{2}[Θ̃_{1}J] = F_{3}F_{2}G_{1} with F_{2} ≤ L_{2}, and Θ̃_{1}[Θ̃_{2}J] = L_{3}H_{2}H_{1} with G_{2} ≤ H_{2}.

Remark 3.3. The condition (c), in the previous lemma, imposed on the operators of the hexagon (3.3) does not come from the braid relations of the operators Θ̃_{i}. As can be seen in the example below, there are operators Θ̃_{1} and Θ̃_{2} which close the hexagon and do not satisfy the conditions in (c).

1 2 4 3  1 3 2 4  2 1 3 4  2 1 3 4 1 2 4 3  1 3 2 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  .

We start to analyse the hexagon (3.3) under the conditions in (c) of the previous lemma. The Knuth class of a key-tableau over a three-letter alphabet as well as any frank word with three columns can be characterized in terms of the shuffling operation. This characterization gives a combinatorial explanation of our hexagons (3.1), (3.2) and (3.3). Indeed by Greene’s theorem [15], the set of all shuffles of the columns of a key-tableau are contained in its Knuth class. However, under certain conditions we have equality.

Theorem 3.4. [6] Let K be a key-tableau with first column A. Then, the Knuth class of K is equal to the set of all shuffles of its columns if and only if each of its column is either an interval of A or is obtained from an interval of A by removing a single letter.

This criterion can be easily applied considering the planar representation of the weight of the key-tableau. For instance K(2, 0, 1, 2, 4, 2, 3) is the shuffle of its columns, since each column in the planar representation of the weight (2, 0, 1, 2, 4, 2, 3),

7 • • •
6 • •
5 • • • •
4 • •
3 •
1 • •
1 2 3 4

has, at most, one gap of size 1. Each column is either an interval of A = {1, 3, 4, 5, 6, 7} or is obtained from an interval of A removing one letter.

Corollary 3.1. The following statements are equivalent:

(a) The Knuth class of a key-tableau over a three-letter alphabet is the set of all shuffles of its columns.

(b) J is a three-column frank word if and only if J has one of the following forms

(I) \begin{matrix}A_{1}^{1} \\ A_{1}^{2} & A_{2}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ; (II) \begin{matrix}& A_{2}^{1} \\ A_{1}^{2} & A_{2}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ; (III) \begin{matrix}& & A_{3}^{1} \\ A_{1}^{2} & & A_{3}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ;

(IV) \begin{matrix}& & A_{3}^{1} \\ & A_{2}^{2} & A_{3}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ; (V) \begin{matrix}A_{1}^{1} \\ A_{1}^{2} & & A_{3}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ; (VI) \begin{matrix}& A_{2}^{1} \\ & A_{2}^{2} & A_{3}^{2} \\ A_{1}^{3} & A_{2}^{3} & A_{3}^{3} \end{matrix} ,

where A_{1}^{3} ≤ A_{2}^{3} ≤ A_{3}^{3}, with |A_{1}^{3}| = |A_{2}^{3}| = |A_{3}^{3}|; A_{i}^{r} ∩ A_{i}^{s} = ∅, for r ≠ s, i = 1, 2, 3, and A_{1}^{2} ≤ A_{2}^{2}, A_{1}^{2} ≤ A_{3}^{2}, A_{2}^{2} ≤ A_{3}^{2}, with |A_{1}^{2}| = |A_{2}^{2}| = |A_{3}^{2}|.

Proof. This follows from the previous theorem. □

Every three-column frank word is a shuffle of row words of length ≤ 3. Clearly the operators Θ̃_{i} acting row words of length ≤ 3 close the hexagons, and satisfy trivially the conditions (c) of Lemma 3.2. In this case, we have indeed Θ̃_{i} = Θ_{i}. But if the columnshape of the frank word is any permutation of (1, 1, 2), the action of the operator Θ_{i} does not always split into an action on row words of length 3 and on row words of length 1.

Let J = \begin{aligned}c^{2} & b^{4} & a^{4} \\ & & a^{2} \end{aligned} be a three-column contre-tableau. We have to distinguish three

situations. Either we have c^{2} ≤ a^{2} < b^{4} ≤ a^{4} or c^{2} ≤ b^{4} ≤ a^{2} < a^{4} or a^{2} < c^{2} ≤ b^{4} ≤ a^{4}.

We say that a hexagon closes in the Knuth class when all elements appearing in the hexagon belong to the same Knuth class.

If c^{2} ≤ a^{2} < b^{4} ≤ a^{4}, there is only one hexagon closing in the Knuth class of J, namely the one defined by Θ̃_{i} = Θ_{i}. It satisfies conditions (c) of Lemma 3.2, but the operators do not split into operators acting on row words. The full collection of hexagons satisfying conditions (c) of Lemma 3.2, in this case, is described as follows. When c^{2} = a^{2}, the following (closing in the Knuth class) is the only hexagon satisfying conditions (c) of Lemma 3.2,

J = \begin{aligned}& b^{4} & a^{4} \\ c^{2} & & a^{2} \end{aligned}  \begin{aligned}& b^{4} & a^{4} \\ c^{2} & a^{2} \end{aligned}  \begin{matrix}b^{4} & & a^{4} \\ c^{2} & a^{2} \end{matrix}  \begin{matrix}b^{4} & & a^{4} \\ c^{2} & a^{2} \end{matrix} \begin{aligned}& b^{4} & a^{4} \\ c^{2} & & a^{2} \end{aligned}  \begin{aligned}& b^{4} & a^{4} \\ c^{2} & a^{2} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  .

When c^{2} < a^{2}, besides the hexagon closing in the Knuth class, there is still another hexagon (obviously not closing in the Knuth class) in the conditions (c) of Lemma 3.2,

J = \begin{aligned}& b^{4} & a^{4} \\ c^{2} & & a^{2} \end{aligned}  \begin{aligned}& b^{4} & a^{4} \\ c^{2} & a^{2} \end{aligned}  \begin{matrix}c^{2} & & a^{4} \\ a^{2} & b^{4} \end{matrix}  \begin{matrix}c^{2} & & a^{4} \\ a^{2} & b^{4} \end{matrix} \begin{aligned}& b^{4} & a^{4} \\ c^{2} & & a^{2} \end{aligned}  \begin{aligned}& b^{4} & a^{4} \\ c^{2} & a^{2} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  .

If c^{2} ≤ b^{4} ≤ a^{2} < a^{4}, we have two hexagons closing in the conditions (c) of Lemma 3.2: one closing in the Knuth class of J, where the operators Θ̃_{i} = Θ_{i} can be split by their action on the row words c^{2} ≤ b^{4} ≤ a^{2} and a^{4} (for convenience J is written as a shuffle of those rows),

J = \begin{aligned}c^{2} & b^{4} & a^{2} \\ & & a^{4} \end{aligned}  \begin{aligned}c^{2} & b^{4} & a^{2} \\ & a^{4} \end{aligned}  \begin{matrix}c^{2} & b^{4} & a^{2} \\ a^{4} \end{matrix}  \begin{matrix}c^{2} & b^{4} & a^{2} \\ a^{4} \end{matrix} \begin{aligned}c^{2} & b^{4} & a^{2} \\ & & a^{4} \end{aligned}  \begin{aligned}c^{2} & b^{4} & a^{2} \\ & a^{4} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  ;

and the other one, when b^{4} < a^{2}, not closing in the Knuth class of J, where the operators Θ̃_{i} are split by their action on the row words c^{2} ≤ b^{4} < a^{4} and a^{2},

J = \begin{aligned}c^{2} & b^{4} & a^{4} \\ & & a^{2} \end{aligned}  \begin{aligned}c^{2} & b^{4} & a^{4} \\ & a^{2} \end{aligned}  \begin{matrix}c^{2} & b^{4} & a^{4} \\ a^{2} \end{matrix}  \begin{matrix}c^{2} & b^{4} & a^{4} \\ a^{2} \end{matrix} \begin{aligned}c^{2} & b^{4} & a^{4} \\ & & a^{2} \end{aligned}  \begin{aligned}c^{2} & b^{4} & a^{4} \\ & a^{2} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  .

If a^{2} < c^{2} ≤ b^{4} ≤ a^{4}, there is only one hexagon satisfying conditions (c) of Lemma 3.2, the one closing in the Knuth class of J, which splits over the rows c^{2} ≤ b^{4} ≤ a^{4} and a^{2}.

Example 3.5. From the discussion above, there are only two hexagons in the conditions

(c) of Lemma 3.2 having the contre-tableau J = 1 2 4 3 as a vertex. The second one

gives the frank words in the Knuth class.

J = 1 2 4 3  1 3 2 4  3 1 2 4  3 1 2 4 1 2 4 3  1 3 2 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  ,

J = 1 2 3 4  1 4 2 3  4 1 2 3  4 1 2 3 1 2 4 3  1 4 2 3^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  .

In the case of the contre-tableau J′ = 1 3 4 2 where 1 < 2 < 3 < 4, we have also

only two hexagons in the conditions (c) of Lemma 3.2. The first hexagon gives the frank words in the Knuth class:

J′ = 1 3 4 2  1 3 2 4  1 3 2 4  1 3 2 4 1 3 4 2  1 2 3 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  ,

J′ = 1 3 4 2  1 3 2 4  1 2 3 4  1 2 3 4 1 3 4 2  1 2 3 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  .

The meaning of Remark 3.3 becomes now clear. As any three-column frank word is a shuffle of rows of length ≤ 3, and frank words with column-shape equals to a permutation of (2, 1, 1), then, given a three-column frank word, it is always possible to construct a hexagon satisfying conditions (c) of Lemma 3.2 and having that frank word as a vertex. Define such a hexagon as a shuffle of those above. Our next main theorem makes this shuffle precise and moreover shows that any hexagon on three-column frank words satisfying conditions (c) of Lemma 3.2 is exactly a shuffle of those hexagons.

Theorem 3.6. Let J = J_{3}J_{2}J_{1} be a three-column contre-tableau. The following assertions are equivalent.

(a) There exist operators Θ̃_{1} and Θ̃_{2} defining the hexagon (3.3) such that Θ̃_{2}[Θ̃_{1}J] = F_{3}F_{2}G_{1} with F_{2} ≤ L_{2}, and Θ̃_{1}[Θ̃_{2}J] = L_{3}H_{2}H_{1} with G_{2} ≤ H_{2}.

(b) The contre-tableau J has a decomposition, as below, giving rise to the hexagon of frank words with the same weight, with T a tableau,

(3.4)

\begin{aligned}& A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ & & A_{2}^{4} & A_{1}^{4} \\ J= & & A_{2}^{3} & A_{1}^{3} \\ & A_{3}^{2} & & A_{1}^{2} \\ & & & A_{1}^{1} \end{aligned}  \begin{aligned}A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ & A_{2}^{4} & A_{1}^{4} \\ & A_{2}^{3} & A_{1}^{3} \\ A_{3}^{2} & A_{1}^{2} \\ & A_{1}^{1} \end{aligned}  \begin{matrix}A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ A_{2}^{4} & & A_{1}^{4} \\ A_{2}^{3} & & A_{1}^{3} \\ A_{3}^{2} & A_{1}^{2} \\ A_{1}^{1} \end{matrix}  \begin{aligned}& A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ & A_{2}^{4} & & A_{1}^{4} \\ T= & A_{2}^{3} & A_{1}^{3} \\ & A_{3}^{2} & A_{1}^{2} \\ & A_{1}^{1} \end{aligned} \begin{aligned}A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ & A_{2}^{4} & A_{1}^{4} \\ A_{2}^{3} & & A_{1}^{3} \\ A_{3}^{2} & & A_{1}^{2} \\ & & A_{1}^{1} \end{aligned}  \begin{aligned}A_{3}^{5} & A_{2}^{5} & A_{1}^{5} \\ & A_{2}^{4} & A_{1}^{4} \\ A_{2}^{3} & A_{1}^{3} \\ A_{3}^{2} & A_{1}^{2} \\ & A_{1}^{1} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  Θ̃_{2}  ,

such that the sets A_{i}^{j} are pairwise disjoint in each column J_{i},

A_{i+1}^{j} ≤ A_{i}^{j}, with |A_{i+1}^{j}| = |A_{i}^{j}|;
A_{3}^{2} ≤ A_{1}^{2} < A_{2}^{4} ≤ A_{1}^{4}, A_{1}^{2} ∩ A_{2}^{4} = ∅, with |A_{3}^{2}| = |A_{1}^{2}| = |A_{2}^{4}| = |A_{1}^{4}|;

and J_{1}∩A_{2}^{5} ⊆ A_{1}^{5}; (J_{1}\A_{1}^{5})∩A_{2}^{4} ⊆ A_{1}^{4}; [J_{1}\(A_{1}^{5}∪A_{1}^{4})]∩A_{2}^{3} ⊆ A_{1}^{3}; [J_{2}∪(A_{1}^{2}∪A_{1}^{1})]∩A_{3}^{2} ⊆ A_{1}^{2}; and [J_{2} ∪ (A_{1}^{2} ∪ A_{1}^{1})] ∩ A_{3}^{5} ⊆ A_{2}^{5}.

Proof. (b) ⇒ (a) By previous corollary, the vertices of the hexagon (3.4) are frank words with the same weight, and clearly satisfy (c) of Lemma 3.2.

(a) ⇒ (b) The frank words J_{3}J_{2}J_{1} and J_{3}G_{2}G_{1} are, respectively, in the conditions (IV) and (II) of Corollary 3.1 and satisfy Θ̃_{1}J_{3}J_{2}J_{1} = J_{3}G_{2}G_{1}. Then

G_{1} ⊆ J_{1}, |G_{1}| = |J_{2}|, J_{2} ≤ G_{1}, J_{1} ∩ J_{2} ⊆ G_{1} and

G_{2} = J_{2} ∪ (J_{1} \ G_{1}), J_{3} ≤ G_{2}. (3.5)

Since the frank word Θ̃_{2}(J_{3}J_{2}J_{1}) = L_{3}L_{2}J_{1} satisfy conditions (III) of Corollary 3.1 we have L_{2} ⊆ J_{2}, |L_{2}| = |J_{3}|, J_{3} ≤ L_{2} ≤ J_{1} J_{2} ∩ J_{3} ⊆ L_{2} and L_{3} = J_{3} ∪ (J_{2} \ L_{2}). Again the frank word F_{3}F_{2}G_{1} = Θ̃_{2}(J_{3}G_{2}G_{1}) satisfy (V) of Corollary 3.1. Then

F_{2} ⊆ G_{2}, |F_{2}| = |J_{3}|, J_{3} ≤ F_{2} ≤ G_{1}, G_{2} ∩ J_{3} ⊆ F_{2} and

F_{3} = J_{3} ∪ (G_{2} \ F_{2}). (3.6)

By (3.5) and (3.6), we have F_{2} ⊆ G_{2} = J_{2} ∪(J_{1} \G_{1}). Thus, we may write F_{2} = A_{2}^{5} ∪A_{1}^{2}, with A_{2}^{5} ⊆ J_{2} and A_{1}^{2} ⊆ J_{1}\G_{1}. Moreover, since J_{3} ≤ F_{2}, we may also write J_{3} = A_{3}^{5}∪A_{3}^{2}, where A_{3}^{5} ≤ A_{2}^{5} e A_{3}^{2} ≤ A_{1}^{2} satisfy |A_{3}^{5}| = |A_{2}^{5}|, |A_{3}^{2}| = |A_{1}^{2}|, G_{2} ∩ A_{3}^{5} ⊆ A_{2}^{5} and G_{2} ∩ A_{3}^{2} ⊆ A_{1}^{2}. We define A_{1}^{1} = J_{1} \ (G_{1} ∪ A_{1}^{2}), therefore J_{1} \ G_{1} = A_{1}^{1} ∪ A_{1}^{2}.

The frank word F_{3}XH_{1} = Θ̃_{1}F_{3}F_{2}G_{1} satisfy (I) of Corollary 3.1. Then

H_{1} ⊆ G_{1}, |H_{1}| = |F_{2}|, F_{2} ≤ H_{1}, F_{2} ∩ G_{1} ⊆ H_{1} and

F_{3} ▷ X = F_{2} ∪ (G_{1} \ H_{1}) ▷ H_{1}. (3.7)

Since F_{2} = A_{2}^{5} ∪ A_{1}^{2} ≤ H_{1}, we can define A_{1}^{5} = min{Z ⊆ H_{1} : |Z| = |A_{2}^{5}| and A_{2}^{5} ≤ Z}, where the minimum is taken with respect to ≤, and A_{1}^{4} = H_{1} \ A_{1}^{5}. As H_{1} ⊆ G_{1}, put A_{1}^{3} = G_{1} \ H_{1}. We have H_{1} = A_{1}^{5} ∪ A_{1}^{4} and X = A_{2}^{5} ∪ A_{1}^{2} ∪ A_{1}^{3}. From F_{2} ≤ H_{1} and the definition of A_{1}^{5}, we get A_{3}^{5} ≤ A_{2}^{5} ≤ A_{1}^{5} and A_{3}^{2} ≤ A_{1}^{2} < A_{1}^{4}, where A_{1}^{2} ∩ A_{1}^{4} = ∅. Note that from (3.5) and (3.7), we obtain J_{1} ∩ A_{2}^{5} ⊆ A_{1}^{5}. By Lemma 3.2

(3.8)  F_{2} ≤ L_{2}.

Now we consider the bottom edges of our hexagon (3.3). Since the frank word L_{3}H_{2}H_{1} = Θ̃_{1}(L_{3}L_{2}J_{1}) satisfy (II) of Corollary 3.1 we have

H_{1} ⊆ J_{1}, |H_{1}| = |L_{2}|, L_{2} ≤ H_{1}, L_{2} ∩ J_{1} ⊆ H_{1} and

L_{3} ≤ H_{2} = L_{2} ∪ (J_{1} \ H_{1}) ▷ H_{1}. (3.9)

By Lemma 3.2, (c), we have

(3.10)  G_{2} ≤ H_{2}.

Finally, since F_{3}XH_{1} = Θ̃_{2}(L_{3}H_{2}H_{1}), we have X ⊆ H_{2}, |X| = |L_{3}|, L_{3} ≤ X, H_{2} ∩L ⊆ X and F_{3} = L_{3} ∪ (H_{2} \ X).

By (3.9) and A_{2}^{5} ∪ A_{1}^{2} ∪ A_{1}^{3} = X_{2} ⊆ H_{2} = L_{2} ∪ A_{1}^{1} ∪ A_{1}^{2} ∪ A_{1}^{3}, we conclude that A_{2}^{5} ⊆ L_{2} ∪ A_{1}^{1}. Since A_{2}^{5} and A_{1}^{1} are disjoint sets, it follows A_{2}^{5} ⊆ L_{2}. Define A_{2}^{4} = L_{2} \ A_{2}^{5} and A_{2}^{3} = J_{2} \ L_{2}. As |L_{2}| = |H_{1}|, we also have |A_{1}^{4}| = |A_{2}^{4}|, |A_{1}^{3}| = |A_{2}^{3}|, (J_{1} \ A_{1}^{5}) ∩ A_{2}^{4} ⊆ A_{1}^{4} and (J_{1} \ (A_{1}^{5} ∪ A_{1}^{4})) ∩ A_{2}^{3} ⊆ A_{1}^{3}. Moreover from the inequality L_{2} ≤ H_{1}, we get A_{2}^{4} ≤ A_{1}^{4}. By (3.8) and (3.5), we get A_{1}^{2} < A_{2}^{4} with A_{1}^{2} ∩ A_{1}^{4} = ∅, and by (3.10), we have A_{2}^{3} ≤ A_{1}^{3}. □

Considering all tableaux of a given shape and weight, this theorem defines an action of the symmetric group on frank words in the union of the Knuth classes of those tableaux. See Example 3.15.

A right (left) key K̃_{+}(T) (K̃_{−}(T)) of the tableau T is the key-tableau of the same shape as T whose j-th column is the first (last) column of any skew-tableau in a hexagon (3.4) with the following property: its first (last) column has the same length as the j-th column of T (cf. [19]).

From the hexagon (3.4) we get, respectively, a right key of T,

K̃_{+}(T) = \begin{matrix}A_{1}^{5} & A_{1}^{5} & A_{1}^{5} \\ A_{1}^{4} & A_{1}^{4} & A_{1}^{4} \\ A_{1}^{3} & A_{1}^{3} \\ A_{1}^{2} \\ A_{1}^{1} \end{matrix},

and a left key of T,

K̃_{−}(T) = \begin{matrix}A_{3}^{5} & A_{3}^{5} & A_{3}^{5} \\ A_{2}^{4} \\ A_{2}^{3} & A_{2}^{3} \\ A_{3}^{2} & A_{3}^{2} & A_{3}^{2} \\ A_{1}^{1} \end{matrix},

with K̃_{+}(T) ≥ K̃_{−}(T).

20 OLGA AZENHAS AND RICARDO MAMEDE

Example 3.7. Below we give two decompositions of the tableau T = 5 4 3 3 2 2 5 leading to different left and right keys. The second hexagon gives the frank words in the Knuth class of T: (3.11)

3 3 5 5 2  2 4  3 3 5 5 2 2 4  3  3 5  5 2 2 4  \begin{matrix}3 & 3 \\ 5 & & 5 & = T \\ 2 & 2 \end{matrix} 3  3 5 5 2  2 4  3 5 5 2 2 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  , K_{+}(T) = 5 4 3 5 2 3 5,

3 3 5 5 2  2 4  3 3 5 5 2 2 4  3  3 5  5 2 2 4  \begin{matrix}3 & & 3 \\ 5 & 5 & & = T \\ 2 & 2 \\ 4 \end{matrix} 3 3 5  5 2  2 4  3 3 5 5 2 2 4^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ_{2}  , K_{+}(T) = 5 4 3 5 2 3 3.

We may now describe the hexagon (3.2). Without loss of generality, we may consider the hexagon (3.4) in simplified form in the sense that the sets A_{i}^{j} are singular,

(3.12) J = \begin{aligned}c^{5} & b^{5} & a^{5} \\ & b^{4} & a^{4} \\ & b^{3} & a^{3} \\ c^{2} & & a^{2} \\ & & a^{1} \end{aligned}  \begin{aligned}c^{5} & b^{5} & a^{5} \\ & b^{4} & a^{4} \\ & b^{3} & a^{3} \\ c^{2} & a^{2} \\ & a^{1} \end{aligned}  \begin{matrix}c^{5} & b^{5} & a^{5} \\ b^{4} & & a^{4} \\ b^{3} & & a^{3} \\ c^{2} & a^{2} \\ a^{1} \end{matrix}  \begin{matrix}c^{5} & b^{5} & a^{5} \\ b^{4} & & a^{4} \\ b^{3} & a^{3} \\ c^{2} & a^{2} \\ a^{1} \end{matrix} \begin{aligned}c^{5} & b^{5} & a^{5} \\ & b^{4} & a^{4} \\ b^{3} & & a^{3} \\ c^{2} & & a^{2} \\ & & a^{1} \end{aligned}  \begin{aligned}c^{5} & b^{5} & a^{5} \\ & b^{4} & a^{4} \\ b^{3} & a^{3} \\ c^{2} & a^{2} \\ & a^{1} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} Θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} Θ̃_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  Θ̃_{2}  ,

with c^{5} ≤ b^{5} ≤ a^{5}, b^{3} ≤ a^{3}, and c^{2} ≤ a^{2} < b^{4} ≤ a^{4}. The contre-tableau J is therefore split into the frank word Y_{1} = c^{2}b^{4}a^{4}a^{2} of shape (1, 1, 2), and the row words X_{2} = c^{5}b^{5}a^{5}, X_{3} = b^{3}a^{3}, and X_{4} = a^{1}. Let X_{1} = c^{2}a^{2}b^{4}a^{4}. We consider the biwords with pairwise distinct biletters

(3.13)

Σ′ = ( \begin{matrix}J_{3} & J_{2} & J_{1} \\ 3^{2} & 2^{3} & 1^{5} \end{matrix} ) ⟷ Π = ( \begin{matrix}c^{2}a^{2}b^{4}a^{4} & c^{5}b^{5}a^{5} & b^{3}a^{3} & a^{1} \\ 3 1 2 1 & 3 2 1 & 2 1 & 1 \end{matrix} ) ⟷ Σ = ( \begin{matrix}(J_{3}J_{2}J_{1}) ↑ \\ w \end{matrix} ) ,

where Π is obtained by sorting the biletters of Σ′, and Σ is obtained by sorting the biletters of Π in weakly increasing rearrangement for the anti-lexicographic order. Since (J_{3}J_{2}J_{1}) ↑ is a shuffle of the words X_{1}, X_{2}, X_{3} and X_{4}, then w is a shuffle of 3121,

321, 21 and 1 such that the biword Σ is a shuffle of ( \begin{matrix}X_{1} \\ 3121 \end{matrix} ), ( \begin{matrix}X_{2} \\ 321 \end{matrix} ), ( \begin{matrix}X_{3} \\ 21 \end{matrix} ) and

( \begin{matrix}X_{4} \\ 1 \end{matrix} ). Therefore the hexagon (3.2) is a “shuffle” of four elementary hexagons,

(3.14)  ^{(\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 2 3 1 \end{matrix})}^{(\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 1 2 1 \end{matrix})} ^{(\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 2 2 1 \end{matrix})}  ^{(\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 2 3 1 \end{matrix})} (\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 1 2 1 \end{matrix})  (\begin{matrix}c^{2}a^{2}b^{4}a^{4} \\ 3 2 2 1 \end{matrix}) θ_{1} θ_{1}  θ_{2} θ_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

(3.15)  ^{(\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix})}^{(\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix})}  ^{(\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix})}  ^{(\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix})} (\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix})  (\begin{matrix}c^{5}a^{5}b^{5} \\ 3 2 1 \end{matrix}) θ_{1} θ_{1}  θ_{2} θ_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

(3.16)  ^{(\begin{matrix}b^{3}a^{3} \\ 3 2 \end{matrix})}^{(\begin{matrix}b^{3}a^{3} \\ 2 1 \end{matrix})} (\begin{matrix}b^{3}a^{3} \\ 3 1 \end{matrix})  (\begin{matrix}b^{3}a^{3} \\ 3 2 \end{matrix})^{(\begin{matrix}b^{3}a^{3} \\ 2 1 \end{matrix})}  ^{(\begin{matrix}b^{3}a^{3} \\ 3 1 \end{matrix})} θ_{1} θ_{1}  θ_{2} θ_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

(3.17)  ^{(\begin{matrix}a^{1} \\ 3 \end{matrix})}^{(\begin{matrix}a^{1} \\ 1 \end{matrix})}  (\begin{matrix}a^{1} \\ 1 \end{matrix})  (\begin{matrix}a^{1} \\ 2 \end{matrix})^{(\begin{matrix}a^{1} \\ 2 \end{matrix})}  ^{(\begin{matrix}a^{1} \\ 3 \end{matrix})} θ_{1} θ_{1}  θ_{2} θ_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  .

A Yamanouchi tableau is a key-tableau whose shape and weight coincide. A Yamanouchi word is a word congruent to a Yamanouchi tableau. By Corollary 3.1, every three-letter Yamanouchi word w is a shuffle of k ≥ 0 words 3121, l_{1} ≥ 0 words 321, l_{2} ≥ 0 words 21 and l_{3} − k ≥ 0 words 1, that, by abuse of notation, we shall write w = shuffle((3121)^{k}, (321)^{l_{1}}, (21)^{l_{2}}, 1^{l_{3}−k}).

Theorem 3.8. The hexagon (3.2) is a “shuffle” of the hexagons defined by the bottom rows of the four hexagons (3.14), (3.15), (3.16), and (3.17) with appropriate multiplicities. That is, there exist a shuffle of k ≥ 0 words 3121, l_{1} ≥ 0 words 321, l_{2} ≥ 0 words 21, and l_{3} − k ≥ 0 words 1, w = shuffle((3121)^{k}, (321)^{l_{1}}, (21)^{l_{2}}, 1^{l_{3}−k}), such that

(a) θ̃_{i}w = shuffle((θ_{i}3121)^{k}, (θ_{i}321)^{l_{1}}, (θ_{i}21)^{l_{2}}, (θ_{i}1)^{l_{3}−k}), i = 1, 2;

(b) θ̃_{i}θ̃_{j}w = shuffle((θ_{i}θ_{j}3121)^{k}, (θ_{i}θ_{j}321)^{l_{1}}, (θ_{i}θ_{j}21)^{l_{2}}, (θ_{i}θ_{j}1)^{l_{3}−k}), 1 ≤ i ≠ j ≤ 2;

(c) θ̃_{1}θ̃_{2}θ̃_{1}w = shuffle((θ_{1}θ_{2}θ_{1}3121)^{k}, (θ_{1}θ_{2}θ_{1}321)^{l_{1}}, (θ_{1}θ_{2}θ_{1}21)^{l_{2}}, (θ_{1}θ_{2}θ_{1}1)^{l_{3}−k}).

Example 3.9. The hexagon (3.11) gives rise to the hexagon below, where the operators are based on non-standard pairing of parentheses

(3.18)  \underline{3} \underline{1} 2 1 ¯1 \underline{2} \underline{1}  \underline{3} \underline{2} 3 2 ¯3 \underline{3} \underline{1} \underline{3} \underline{2} 3 2 ¯2 \underline{2} \underline{1} \underline{3} \underline{1} 3 1 ¯1 \underline{2} \underline{1}  \underline{3} \underline{2} 3 1 ¯3 \underline{3} \underline{1} \underline{3} \underline{2} 2 1 ¯2 \underline{2} \underline{1}  θ̃_{1} θ_{1}  θ_{2} θ̃_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

(the barred letters indicate the subwords 3121 and 1 in the shuffle).

Remark 3.10. The following example is the translation of Remark 3.3 to hexagon (3.2). The hexagon

3211 3221  3321 3321 3211  3221^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} θ̃_{1}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} θ̃_{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} θ̃_{1}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} θ̃_{2}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} θ̃_{2}

is not a shuffle of the two hexagons (3.15) and (3.17).

Next we show that the family of actions of 𝔖_{3} defined by the operators θ̃_{i} (Θ̃_{i}), i = 1, 2, based on shuffle decompositions of a three-letter Yamanouchi word w (threecolumn tableau) as shown in the previous theorems, includes the action defined by the operators θ_{i} (Θ_{i}), i = 1, 2. This is achieved in the following algorithm, where a special shuffle decomposition for a three-letter Yamanouchi word w is exhibited. Using (3.13), it follows that the hexagon (3.4) contains, in particular, the action defined by the operators Θ_{i}. That is, the Lascoux–Schützenberger action of the symmetric group: on frank words is obtained from a particular shuffle decomposition of a three-column tableau into rows of length ≤ 3, and tableaux of column-shape (2, 1, 1); and on words congruent to key-tableaux is obtained from a shuffle decomposition of a three-letter Yamanouchi word into words 3121 and column-words 321, 21, 1.

Let w = w_{1} ⋯ w_{l} be a word of length l. We denote by w_{|A} the subword of w obtained by suppressing the letters not in A. If X ⊆ [l], then w|X is the subword of w defined by the letters of w in positions X. If X, Y ⊆ [l] with X ∩ Y = ∅, then w|(X, Y) is the shuffle of the subwords w|X and w|Y defined by the letters of w in positions X ∪Y. By induction, we define w|(X_{1}, …, X_{k}), for any k ≥ 0, putting the empty word for k = 0. Given the positive integers i ≤ j, we put [i, j] for the set {i, i + 1, …, j}.

Algorithm 3.11. Let w be a Yamanouchi word over a three-letter alphabet. Our algorithm is consists of three steps.

Step 1. Consider the subword w_{|{2,1}} and bracket every factor 21 of w_{|{2,1}}. The letters which are not bracketed constitute a subword of w_{|{2,1}}. Then bracket every factor 21 of this subword. Again, the letters which are not bracketed constitute a subword. Continue this procedure until it stops, that is, until we get a word consisting of l_{1} non-bracketed letters 1′s in w. This bracketing process enables us to decompose w as

(3.19)  w|(I_{1}, …, I_{l_{3}+l_{2}}, J_{1}, …, J_{l_{3}}, K_{1}, …, K_{l_{1}}),

where w|I_{l} = 21, l ∈ [l_{3} + l_{2}], w|J_{l} = 3, l ∈ [l_{3}], and w|K_{l} = 1, l ∈ [l_{1}].

Step 2. Let w′ be the subword of w obtained by removing all letters 1 belonging to the factors w|I_{l}, for all l ∈ [l_{3} +l_{2}]. As in the previous step, we bracket all the successive factors 32 and 31 of w′. We get a new decomposition (3.19), by making the unions of k sets J_{l} with k sets K_{l}, for some integer 0 ≤ q ≤ min{l_{3}, l_{1}}, and making the unions of the remaining l_{3} − q sets J_{l} with l_{3} − q sets I_{l}:

w|(F_{1}, …, F_{q}, G_{1}, …, G_{l_{3}−q}, I_{1}, …, I_{l_{2}+q}, K_{1}, …, K_{l_{1}−q}),

where w|F_{l} = 31, l ∈ [q], w|G_{l} = 321, l ∈ [l_{3} − q], w|I_{l} = 21, l ∈ [l_{2} + q], and w|K_{l} = 1, l ∈ [l_{1} − q] (reordering the sets I_{i}’s, J_{j}’s and K_{l}’s in (3.19) if necessary).

Step 3. Finally, let w′′ be the subword of w obtained by removing the subwords w|G_{l} = 321 and w|K_{l} = 1, for all l ≥ 1. As before, we bracket all the successive factors 3121 of w′′. This operator consists of the union of the q sets F_{l} with q sets I_{l}. The decomposition of w obtained in this way, is denoted by w|(I_{1}^{∗}, …, I_{l_{3}+l_{2}+l_{1}−q}^{∗}), where w|I_{l}^{∗} = 3121, l ∈ [q], w|I_{l}^{∗} = 321, l ∈ [q + 1, l_{3}], w|I_{l}^{∗} = 21, l ∈ [l_{3} + 1, l_{3} + l_{2}], and w|I_{l}^{∗} = 1, l ∈ [l_{3} + l_{2} + 1, l_{3} + l_{2} + l_{1} − q].

The next example illustrates the application of the previous algorithm.

Example 3.12. Let w = 33121121 ≡ K(4, 2, 2). Following the first step of Algorithm 3.11, we bracket all the successive factors 21 of w_{|{1,2}}, that is, 331(21)1(21), obtaining in this way the decomposition

w = w|({4, 5}, {7, 8}, {1}, {2}, {3}, {6}),

where w|{4, 5} = w|{7, 8} = 21, w|{1} = w|{2} = 3 and w|{3} = w|{6} = 1. Next, let w′ = 3312 − 12− (where − indicates the place of the suppressed letters of w) be the subword of w obtained by removing the letters 1 belonging to w|{4, 5} and w|{7, 8}, and bracket all the successive factors 31 and 32 of w′. Thus, we have w′ = 3(31)2−12−, with the letters 3 and 1 belonging to w|{2} and w|{3}, respectively; and then, we have w′_{1} = (3 − −2) − 12−, with the letters 3 and 2 of this factor belonging to w|{1} and w|{4, 5}, respectively. Then, we get the decomposition

w = w|({1, 4, 5}, {7, 8}, {2, 3}, {6}),

with w|{1, 4, 5} = 321, w|{7, 8} = 21, w|{2, 3} = 31 and w|{6} = 1. Finally, let w′′ = −31−−−21 be the subword of w obtained by removing the subwords w|({1, 4, 5} = 321 and w|{6} = 1. This word has only one factor 3121 and thus we get the decomposition

w = w|(\{2, 3, 7, 8\}^{∗}, \{1, 4, 5\}^{∗}, \{6\}^{∗}) = \underline{3} \overline{3 1} \underline{2 1} 1 \overline{2 1},

where the underlined letters define 3121, the overlined letters define 321 and the remaining letter defines the shuffle component 1. It is easy to check that the parentheses matching operations induced by this decomposition are the standard ones:

(3.20)  \underline{3} \overline{3 2} \underline{2 1} 3 \overline{3 1} \underline{3} \overline{3 1} \underline{2 1} 1 \overline{2 1} \underline{3} \overline{3 2} \underline{2 1} 2 \overline{2 1}  \underline{3} \overline{3 2} \underline{2 1} 3 \overline{3 1} \underline{3} \overline{3 1} \underline{2 1} 1 \overline{2 1}  \underline{3} \overline{3 2} \underline{2 1} 2 \overline{2 1} θ_{1} θ_{1}  θ_{2} θ_{2}  θ_{2} θ_{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  .

Theorem 3.13. Let w be a Yamanouchi word over a three-letter alphabet and consider the decomposition w|(I_{1}^{∗}, …, I_{q}^{∗}) given by Algorithm 3.11. Then, for each ij ⋯ k ∈ [2]^{∗}, θ_{i}θ_{j} ⋯ θ_{k}(w) = w^{ij···k} satisfy w^{ij···k}|I_{l} = θ_{i}θ_{j} ⋯ θ_{k}(w|I_{l}), for all l = 1, …, q.

Proof. By the construction of w|(I_{1}^{∗}, …, I_{q}^{∗}), it is clear that θ_{i}θ_{j} ⋯ θ_{k}(w) = w^{ij···k}, for ij ⋯ k ∈ {1, 21, 121}. For the computation of θ_{2}(w), we must match all successive factors 32 of w_{|{2,3}}, until we get the subword 2^{l_{2}}, for some non-negative integer l_{2}. Each matched pair 32 belongs either to a component 321, or 3121, while the letters 2 of the subword 2^{l_{2}} belong to components 21. The word θ_{2}(w) is then obtained by replacing in w the subword 2^{l_{2}} by 3^{l_{2}}. Since θ_{2}(3121) = 3121, θ_{2}(321) = 321 and θ_{2}(21) = 31, it follows that θ_{2}(w) = w^{2}.

Finally, consider the subword θ_{1}θ_{2}(w), obtained by matching the successive factors 21 of θ_{2}(w)_{|{1,2}}. By the construction of w|(I_{1}^{∗}, …, I_{q}^{∗}), each one of the matched pairs 21 belongs either to a component 321 or 3121. On the other hand, each letter of the subword 1^{l_{2}+l_{1}}, obtained by removing the matched pairs, is itself a component or it represents the leftmost letter 1 of a component 3121. Since θ_{1}θ_{2}(3121) = 3221 and θ_{1}θ_{2}(1) = 2, we get θ_{1}θ_{2}(w) = w^{12}. □

Finally, our construction in Theorem 3.6 has a matrix interpretation.

Theorem 3.14. [5] To each hexagon (3.4) it corresponds a hexagon (3.1). That is, given a hexagon (3.4), ∃ U ∼ I such that, for some partition α, {Δ_{α}UK(σβ) : σ ∈ 𝔖_{3}} is a hexagon whose indexing-set frank words are those of (3.4).

Next we give an example of our construction in Theorems 3.6 and 3.8 in the case of 𝔖_{4}.

Example 3.15. Dual permutahedra in 𝔖_{4} generated by variants of jeu de taquin and non-standard reflection crystal operators.

Consider the Yamanouchi word w = 4312211 ∈ [4]^{∗} and the contre-tableau J =

1 1 32 451. The biwords Σ = (\begin{matrix}1112345 \\ 4312211 \end{matrix}) and Σ′ = (\begin{matrix}1 1 32 451 \\ 4 3 22 111 \end{matrix}) correspond by the

dual RSK to the pair (K, P), with K = 4321 21 1 the Yamanouchi tableau of shape (3, 2, 1, 1), and P = 321 51 1 4.

The vertices of the following permutahedron in 𝔖_{4} contains the contre-tableau J = 1 1 32 541 ≡ P = 321 51 1 4 and the tableau T = 321 41 1 5 of the same shape and weight as P. The remaining ones are frank words either in the Knuth classes of P or T.

(3.21)

[figure: 1135; 24; 1; 113; 25; 14; 113; 25; 14; 1135; 24; 1; 13; 2; 115; 4; 13; 2; 115; 4; 3; 2; 1115; 4; 3; 2; 1115; 4; 13; 125; 4; 1; 134; 12; 15; 13; 24; 115; 3; 2; 114; 15; 3; 24; 1115; 3; 114; 2; 15; 34; 12; 115; 3; 1125; 4; 1; 3; 24; 1115; 3; 2; 114; 15; ·; ·; ·; ·; ·; 34; 12; 115; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 13; 24; 115; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 3; 114; 2; 15; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 134; 12; 15; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 3; 1125; 4; 1; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 13; 125; 4; 1; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

The vertices of the corresponding dual permutahedron in 𝔖_{4} contain the Yamanouchi words 4312211 ≡ K(3, 2, 1, 1) and 4324431 ≡ K(1, 1, 2, 3) with Q-symbols

Q = 6 3 2 7 1 4 5 = (std P)^{τ} and Q′ = 7 3 2 6 1 4 5 = (std T)^{τ}, respectively.

The Q-symbols of the remain ones are either Q or Q′.

(3.22)

[figure: 4312211; 4322211; 4322211; 4312211; 4323311; 4323311; 4324411; 4324411; 4312311; 4322321; 4323321; 4324421; 4324431; 4322421; 4323431; 4312411; 4324431; 4324421; ·; ·; ·; ·; ·; 4323431; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 4323321; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 4322421; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 4322321; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 4312411; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; 4312311; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·]

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CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal

E-mail address: oazenhas@mat.uc.pt

CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal

E-mail address: mamede@mat.uc.pt