2.1 Biwords and matrices of integers

We introduce the alphabet $\mathcal {A}_n=\{1,\dots ,n\}$ , and we denote by $\mathcal {A}_n^*$ the set of word of finite length in $\mathcal {A}_n$ . The length of a word p is denoted by $\ell (p)$ , while its content is recorded by an array $\gamma =(\gamma _1, \dots , \gamma _n)$ , where $\gamma _i$ equals the multiplicity of i in p.

Given two natural numbers $n,m$ we denote by $\mathbb {A}_{n,m}$ the set of biwords in the alphabets $\mathcal {A}_n, \mathcal {A}_m$ . A biword $\pi \in \mathbb {A}_{n,m}$ is an array of pairs $\left (\begin {smallmatrix} q_1 & q_2 & \cdots & q_k \\p_1 &p_2 &\cdots &p_k \end {smallmatrix}\right )$ , $k\in \mathbb {N}$ , where $q_i\in \mathcal {A}_m$ , $p_i\in \mathcal {A}_n$ and whose columns are ordered lexicographically. This means that for all i we have $q_i \le q_{i+1}$ and whenever $q_i=q_{i+1}$ , then $p_i \le p_{i+1}$ . Clearly, words $p_1 \dots p_k$ are particular cases of biwords, obtained setting $q_i=i$ . Permutations $\sigma \in \mathcal {S}_n$ are also particular cases of biwords where we set $q_i=i$ and $p_i=\sigma _i$ .

Given $\mathcal {A}_n,\mathcal {A}_m$ we consider weighted biwords in these alphabets and we denote their set by $\overline {\mathbb {A}}_{n,m}$ . Elements of $\overline {\mathbb {A}}_{n,m}$ are arrays of tripletsFootnote ⁴

(2.1) $$ \begin{align} \overline{\pi} = \left(\begin{matrix} q_1 & q_2 & \cdots & q_k \\ p_1 & p_2 & \cdots & p_k \\w_1 & w_2 &\cdots &w_k \end{matrix}\right), \end{align} $$

where again $q_i\in \mathcal {A}_m$ , $p_i \in \mathcal {A}_n$ and weights $w_i \in \mathbb {Z}$ . Columns of a weighted biword are arranged so that the biword composed by $q,p$ is lexicographically ordered, with top entries taking precedence, while if $q_i=q_{i+1}$ and $p_i=p_{i+1}$ , then $w_i \ge w_{i+1}$ . The total weight of a weighted biword $\overline {\pi }$ is the sum of the $w_i$ entries and we denote it by $\mathrm {wt}(\overline {\pi })$ , that is, $\mathrm {wt}(\overline {\pi }) = \sum _i w_i$ . In this text, weighted biwords will always be denoted by overlined Greek letters $\overline {\pi },\overline {\sigma },\overline {\xi },\dots $ so to distinguish them from biwords, whose symbols are never overlined.

For later use, we also present an alternative format to express a weighted biword, which we call timetable ordering. Given $\overline {\pi }$ its timetable ordering is the array consisting of the same triplets of $\overline {\pi }$ arranged in such a way that and in case then and if also then . Examples of a weighted biword and of its timetable ordering are

(2.2)

Particular cases of weighted biwords are weighted words, where we assume $q_i=i$ , or weighted permutations, where $q_i=i$ and $p_1,\cdots , p_k$ form a permutation of $\{ 1, \dots , k\}$ . We also use the notion of partial (weighted) permutations, that are weighted biwords where each q and p rows present no repetitions. The set of weighted biwords where all weights are nonnegative integers will be important to us and is denoted with $\overline {\mathbb {A}}_{n,m}^+$ .

Biwords of $\mathbb {A}_{n,m}$ are in natural bijection with the set of rectangular matrices with nonnegative integral entries

(2.3)

Such correspondence is realized assigning to $\pi $ the matrix m with elements

(2.4) $$ \begin{align} M_{i,j}=\# \text{ of } \binom{j}{i} \text{ in } \pi. \end{align} $$

Analogously, weighted biwords of $\overline {\mathbb {A}}_{n,m}$ are in correspondence with rectangular matrices whose elements are eventually vanishing sequences of nonnegative integers

(2.5) $$ \begin{align} \overline{\mathbb{M}}_{n\times m} = \{ (\overline{M}_{i,j}:\mathbb{Z} \to \mathbb{N}_0 : 1\le i \le n, 1 \le j \le m) : \overline{M}_{i,j}(k) = 0 \text{ for } |k| \gg 0 \}. \end{align} $$

Also in this case to a weighted biword $\overline {\pi }$ , we assign the matrix $\overline {M}$ defined by

(2.6) $$ \begin{align} \overline{M}_{i,j}(k) = \# \text{ of } \left( \begin{matrix} j\\i\\k \end{matrix} \right) \text{ in } \overline{\pi}. \end{align} $$

As earlier, we will always denote matrices of $\overline {\mathbb {M}}_{n \times m}$ by overlined capital letters to distinguish them from those of $\mathbb {M}_{n \times m}$ . The subset of $\overline {\mathbb {M}}_{n \times m}$ in bijection with nonnegatively weighted biwords $\overline {\mathbb {A}}_{n,m}^+$ is denoted by $\overline {\mathbb {M}}_{n \times m}^+$ . The weight of a matrix is defined as

$$ \begin{align*} \mathrm{wt}(\overline{M}) = \sum_{k \in\mathbb{Z}}k \sum_{i=1}^n \sum_{j=1}^m M_{i,j}(k) \end{align*} $$

so that under correspondence (2.6) we have $\mathrm {wt}(\overline {M}) = \mathrm {wt}(\overline {\pi })$ . We will represent matrices $\overline {M}\in \overline {\mathbb {M}}_{n \times m}$ , with a slight abuse of notation, as compactly supported maps $\overline {M} : \{1, \dots , m\} \times \mathbb {Z} \to \mathbb {N}_0$ via the identification

(2.7) $$ \begin{align} \overline{M}(j,i-kn) = \overline{M}_{i,j}(k), \qquad \text{for all} \qquad i\in \mathcal{A}_n,\, j\in \mathcal{A}_m, \, k\in \mathbb{Z}. \end{align} $$

Given two weighted biwords $\overline {\pi },\overline {\pi }'$ , we consider their disjoint union formed taking all columns of $\overline {\pi }$ and $\overline {\pi }'$ and rearranging them in the correct order. In case $\overline {M},\overline {M}'$ are the matrices corresponding to $\overline {\pi },\overline {\pi }'$ , then naturally $\overline {M} + \overline {M}'$ is the matrix associated to their union.

Throughout the paper, we will consider a number of operations on biwords and most of times these will have a nice description in the language of matrices. For instance, if $\overline {M}$ is the matrix corresponding to a weighted biword $\overline {\pi }$ , then to its transpose $\overline {M}^T$ it will correspond a biword $\overline {\pi }^{-1}$ obtained from $\overline {\pi }$ swapping the p and the q rows and rearranging the result in the prescribed order. This notation is standard and is justified by the fact that when $\overline {\pi }$ is a permutation $\overline {\pi }^{-1}$ is its inverse in the symmetric group.

2.2 Partitions and Young diagrams

A partition $\lambda =(\lambda _1, \lambda _2,\cdots )$ is a weakly decreasing sequence of integers $\lambda _i$ eventually being zero. The number of nonzero parts of $\lambda $ is its length, and it is denoted by $\ell (\lambda )$ . We say that $\lambda $ partitions k if $| \lambda | = \lambda _1 + \lambda _2 + \cdots = k$ and sometimes we write $\lambda \vdash k$ . The multiplicative notation $\lambda =1^{m_1(\lambda )} 2^{m_2(\lambda )}\cdots $ is often used and $m_i(\lambda )$ denotes the multiplicity of i in $\lambda $ . The set of all partitions is denoted by $\mathbb {Y}$ . Sometimes, we will refer to arrays $\varkappa =(\varkappa _1,\dots ,\varkappa _N) \in \mathbb {N}_0^{N}$ of N nonnegative integers as compositions. Given a composition $\varkappa \in \mathbb {N}_0^{N}$ , we denote by $\varkappa ^+$ the unique partition that can be generated permuting elements of $\varkappa $ .

Partitions are identified by their Young diagrams, and we will freely interchange these two notions. Viewing the plane $\mathbb {Z} \times \mathbb {Z}$ with the vertical coordinate increasing downward, the Young diagram of $\lambda $ is the collection of cells $(c,r)$ with $1 \le r \le \ell (\lambda )$ and $1\le c \le \lambda _r$ . Reflecting the Young diagram of $\lambda $ with respect to the main diagonal, we obtain the transposed partition $\lambda '$ with parts $\lambda _i^{\prime } = \# \{ j: \lambda _j \ge i \}$ . Given two partitions $\mu , \lambda $ , we write $\mu \subseteq \lambda $ if $\mu _i \le \lambda _i$ for all i or equivalently if their Young diagrams are encapsulated. When $\mu \subseteq \lambda $ , we define the skew Young diagram $\lambda /\mu $ consisting of all all cells in $\lambda $ but not in $\mu $ . The number of cells of $\lambda /\mu $ is denoted with $|\lambda /\mu |$ .

As hinted in the introduction, for the discussion in this paper, it is essential to allow Young diagrams to have rows at nonpositive coordinates. For this, we define the upward translation $\mathcal {T}_{-i} : (c,r) \mapsto (c,r-i)$ , for any $i\in \mathbb {N}_0$ and the set of generalized Young diagrams $\mathbb {Y}_{-i} = \mathcal {T}_{-i} (\mathbb {Y})$ . A generalized Young diagram $\lambda \in \mathbb {Y}_{-i}$ is associated to its generalized partition $(\lambda _{1-i},\lambda _{2-i},\dots )$ . The notion of skew diagrams is defined as always: If $\mu ,\lambda \in \mathbb {Y}_{-i}$ , then $\lambda /\mu $ is the set of cells in $\lambda $ but not in $\mu $ . When drawing generalized Young diagrams we will color cells at nonpositive rows in gray to give a reference. We report an example of a skew Young diagram and a generalized one obtained translating it

(2.8)

At times, we will need to distinguish generalized Young diagrams from nongeneralized ones, that in these circumstances will be called classical. Statements and constructions reported in this text often apply the same to classical Young diagrams or to generalized Young diagrams and unless required, we will not stress the difference. Nevertheless, we point out that not every operation defined on classical Young diagrams is possible in the generalized case: For instance, the notion of transposition $\lambda '$ is only defined if $\lambda \in \mathbb {Y}$ .

Given a classical Young diagram $\mu $ , its rectangular decomposition is given by indices $0=R_0<R_1<\cdots < R_N=\mu _1$ such that

(2.9) $$ \begin{align} \mu_{R_{i-1}+1}^{\prime} = \cdots = \mu_{R_i}^{\prime}> \mu_{R_i+1}^{\prime}, \end{align} $$

for $i=1,\dots ,N$ ; see Figure 3. When using the notion of rectangular decomposition, we will denote by $r_i=R_{i-1}- R_{i}$ the base of each rectangle of $\mu $ .

Figure 3 $R_i$ ’s and $r_i$ ’s for a partition $\mu $ .

2.3 Young tableaux

A Young tableau, or simply a tableau, T is a filling of cells of a Young diagram with natural numbers. The label assigned to a specific cell $(c,r) \in \lambda $ is indicated with $T(c,r)$ and in this case $\lambda $ is called shape of T. If a cell has label $\ell $ we call that an $\ell $ -cell. The content of a tableau T is recorded by an array $\gamma =(\gamma _1,\gamma _2,\dots )$ , where $\gamma _i = \# i$ -cells in T. We will mainly deal with two types of tableaux. Semistandard tableaux have entries strictly increasing column-wise and weakly increasing row-wise. When their labels range in the alphabet $\mathcal {A}_n$ and the shape $\lambda / \rho $ is fixed, we denote their set by $SST(\lambda /\rho ,n)$ . In such cases, we call $\lambda $ the external shape and $\rho $ the empty shape. It could happen that the shape $\lambda /\rho $ of T is a generalized skew Young diagram, as in the examples in Section 1.2, and in such cases T is a generalized semistandard tableau. As before, we will not stress the generalized property unless required. A particular class of semistandard tableaux is that of standard tableaux, defined by the property of having content $\gamma _i=1$ for all $i=1,\dots , |\lambda / \rho |$ . Their set is denoted by $ST(\lambda /\rho )$ .

The other class of tableaux, which will play an important role in this paper, is that of vertically strict tableaux that have labels strictly increasing column-wise and no additional conditions. These are sometimes called ‘column strict fillings’ [Reference Lenart and Schilling58] or when entries have no repetitions ‘column tabloids’ [Reference Sagan74]. Shapes of vertically strict tableaux will always be straight (i.e., nonskew) classical Young diagrams $\mu \in \mathbb {Y}$ , and we denote their set by $VST(\mu ,n)$ when entries range in the alphabet $\mathcal {A}_n$ . Examples of semistandard and vertically strict tableaux are

(2.10)

with content, respectively, equal to $(2,2,2,1,2)$ and $(2,2,3,1,2)$ .

Given a tableau T, we define its row reading word $\pi ^{\mathrm {row}}_T$ concatenating rows of T starting from the last. Alternatively, the column reading word $\pi ^{\mathrm {col}}_T$ is formed reading entries of T column by column from the last row up and from left to right. For instance, if T is the semistandard tableau in equation (2.10) we have

(2.11) $$ \begin{align} \pi^{\mathrm{row}}_T = 1\,\,2\,\,5\,\,1\,\,3\,\,3\,\,5\,\,2\,\,4 \qquad \textrm{and} \qquad \pi^{\mathrm{col}}_T = 1\,\,2\,\,1\,\,5\,\,3\,\,3\,\,2\,\,5\,\,4. \end{align} $$

2.4 Kernels of tableaux

We now define a useful statistic of a semistandard Young tableau of classical shape.

For example, if

(2.12)

In fact, shifting the second and first row of P, respectively, one and two cells to the left we obtain the semistandard tableau of equation (2.10), which cannot be ‘squeezed’ anymore.

In order to describe more precisely the partition $\ker (P)$ , we introduce the notion of overlap of two weakly increasing words $A,B$ . This is defined as $\ell (B)$ minus the size of the empty shape of the minimal semistandard tableaux having first and second row content given by A and B, respectively. In formulas, we have

(2.13) $$ \begin{align} \mathrm{ov}(A,B) = \max_{L \in \{0,\dots, \ell(A) \wedge \ell(B) \}} \left\{ L : B_{\ell(B)-L+i}> A_{i}, \text{ for all } i=1,\dots,L \right\}. \end{align} $$

For instance, for $A=1 \,\, 3 \,\, 3 \,\, 5$ and $B= 1 \,\, 2 \,\, 2 \,\, 3 \,\, 4$ we have $\mathrm {ov}(A,B) = 2$ since the minimal semistandard tableaux with first row A and second row B is

(2.14)

Recording the j-th row of a tableau $P \in SST(\lambda /\rho ,n)$ in a weakly increasing word $p^{(j)}$ of length $\theta _j=\lambda _j-\rho _j$ and setting $\varkappa =\ker (P)$ , we have

(2.15) $$ \begin{align} \varkappa_j - \varkappa_{j+1} = \rho_j - \lambda_{j+1} + \mathrm{ov}(p^{(j)}, p^{(j+1)}). \end{align} $$

Given a pair $(P,Q)$ of semistandard tableaux with the same shape, we can define $\ker (P,Q)$ in the same way as in Proposition 2.1. If $\nu =\ker (P,Q)$ , then $\nu _j-\nu _{j+1}$ is the maximal amount of cells we can shift the first j rows of P and Q simultaneously to the left without breaking the semistandard property. In this case, if $\varkappa =\ker (P), \kappa =\ker (Q)$ and $\nu =\ker (P,Q)$ . then it is clear that for all $j=1,2\dots $ , we have

(2.16) $$ \begin{align} \nu_{j}-\nu_{j+1} = \min \{ \varkappa_{j} -\varkappa_{j+1} , \kappa_{j} -\kappa_{j+1} \}. \end{align} $$

2.5 Row coordinate parameterization

To any generalized semistandard tableau $P\in SST (\lambda /\rho , n)$ , we can assign its row-coordinate matrix $\alpha = \mathrm {rc}(P)$ , defined by

(2.17) $$ \begin{align} \alpha_{i,j} = \# \text{ of } i\text{-cells at row } j \text{ of } P. \end{align} $$

The set of such infinite matrices is

(2.18)

Such encoding of tableaux was defined already in [Reference Danilov and Koshevoi22]. In case a tableau P is standard, we condense in an array information contained in the row-coordinate matrix. Define the row-coordinate array of a standard tableaux P as

(2.19)

We will abuse of the notation and write rather than $\mathrm {rc}(P)= \alpha \in \mathbb {M}_{n \times \infty }$ when it is clear from the context that P is standard.

The map $\mathrm {rc}:P \mapsto \alpha $ is not bijective since shifting rows of P laterally does not change its row-coordinate matrix. It can nevertheless be refined into a bijection recording in a certain way relative positions of rows of P. We do so in the next definition, where, for the sake of a simpler description, we only consider the case of tableaux with classical shape. We will use the notation $\mathrm {rc}(P,Q) = (\mathrm {rc}(P), \mathrm {rc}(Q))$ .

In the definition above, we have assumed that tableaux $P,Q$ have the same classical shape $\lambda /\rho $ and that their labels belong to the same alphabet $\mathcal {A}_n$ . This forces their row-coordinate matrices to belong to the set

(2.20) $$ \begin{align} \mathcal{M}_n^+ = \left\{ (\alpha,\beta) \in \mathbb{M}_{n \times \infty}^+ \times \mathbb{M}_{n \times \infty}^+ : \sum_{i=1}^n ( \alpha_{i,j} - \beta_{i,j} )= 0 \text{ for all } j\in \mathbb{Z} \right\}, \end{align} $$

where $\mathbb {M}_{n \times \infty }^+$ is the subspace of $\mathbb {M}_{n \times \infty }$ of matrices $\alpha $ such that $\alpha _{i,j} = 0$ if $j \le 0$ .

Proof. We need to construct the inverse map $(\alpha ,\beta ;\nu ) \to (P,Q)$ . For this, define weakly increasing words $p^{(j)},q^{(j)}$ as

(2.21) $$ \begin{align} p^{(j)} = 1^{\alpha_{1,j}} 2^{\alpha_{2,j}} \cdots, \qquad q^{(j)} = 1^{\beta_{1,j}} 2^{\beta_{2,j}} \cdots. \end{align} $$

Since $(\alpha ,\beta ) \in \mathcal {M}_n^+$ , $p^{(j)},q^{(j)}$ have the same length denoted by $\theta _j$ . Define also partition $\eta $ through relations

(2.22) $$ \begin{align} \eta_i-\eta_{i+1} = \theta_{j+1} - \min \left\{ \mathrm{ov}(p^{(j)}, p^{(j+1)}) , \mathrm{ov}(q^{(j)}, q^{(j+1)} ) \right\}. \end{align} $$

Then $P,Q$ are the tableaux of shape $\lambda /\rho $ with $\lambda = \eta + \theta + \nu $ and $\rho = \eta + \nu $ and with j-th rows given by words $p^{(j)},q^{(j)}$ . It is straightforward to check that maps $(P,Q)\mapsto (\alpha ,\beta ;\nu )$ and $(\alpha ,\beta ;\nu ) \mapsto (P,Q)$ are mutual inverses.

Example 2.4. Consider the pair of semistandard tableaux

(2.23)

Then we have , where

(2.24)

For later use, we also introduce the set

(2.25)

consisting on all pairs that are row-coordinate arrays of pairs of standard tableaux. At times in the text, we will also use sets $\mathcal {M}_n$ , $\mathcal {Z}_n$ defined, respectively, as in equation (2.20), replacing $\mathbb {M}_{n\times \infty }^{+}$ by $\mathbb {M}_{n\times \infty }$ and equation (2.25) replacing $\mathbb {N}$ by $\mathbb {Z}$ .

2.6 Standardization

We define the operation of standardization [Reference Schützenberger78] of semistandard tableaux

(2.26) $$ \begin{align} \mathrm{std} : P \in SST(\lambda / \rho , n) \mapsto P' \in ST(\lambda / \rho). \end{align} $$

Let $\gamma = (\gamma _1 ,\dots , \gamma _n)$ be the content of tableau P, and define $\Gamma _i=\gamma _1 + \cdots + \gamma _i$ for $i=1,\dots ,n$ , where $\Gamma _0=0$ by convention. Then cells of $P'=\mathrm {std}(P)$ are labeled replacing i-cells of P, from the leftmost to the right with $\Gamma _{i-1} + 1 , \dots , \Gamma _i$ . For instance, we have

(2.27)

It is clear that, remembering the content $\gamma $ of the original tableaux P, one can recover P from its standardization $P'$ .

We present the analog of standardization in the language of matrices. Rows of matrices in $\mathbb {M}_{n\times \infty }$ are compactly supported infinite arrays of nonnegative integers, and we denote their set by

(2.28) $$ \begin{align} \mathbb{V}=\bigg\{ (v_j)_{j\in \mathbb{Z}}: v_j \in \mathbb{N}_0 \text{ and } |v| = \sum_{j\in \mathbb{Z}} v_j < +\infty \bigg\}. \end{align} $$

We will write elements $v \in \mathbb {V}$ via the expansion $v= \sum _{k \in \mathbb {Z}} v_k \mathbb {e}_k $ , where $\mathbb {e}_k$ is the standard basis of the infinite-dimensional vector space $\bigoplus _{k\in \mathbb {Z}} \mathbb {C}$ . Introducing the Weyl chamber

(2.29)

we define the natural correspondence $\mathbb {W}^k \leftrightarrow \{ v \in \mathbb {V} : |v| = k\},$ by the invertible mapping

(2.30)

Given $\alpha \in \mathbb {M}_{n \times \infty }$ and denoting its rows by $\alpha _1,\alpha _2 ,\dots $ , we define the standardization as the array

(2.31)

obtained joining smaller arrays such that , under correspondence (2.30). One can check that standardization of matrices is compatible with the row-coordinate parameterization, or in other words

(2.32) $$ \begin{align} \mathrm{rc} \circ \mathrm{std} (P) = \mathrm{std} \circ \mathrm{rc} (P) \end{align} $$

for all semistandard tableaux P. An example of such commutation relation can be observed considering the semistandard tableau P on the left-hand side of equation (2.27), whose row-coordinate matrix $\alpha =\mathrm {rc}(P)$ was reported in Proposition 2.4. Then we see that both $\mathrm {std}(\alpha )$ and $\mathrm {rc}(\mathrm {std}(P))$ give as a result the array .

3.1 Skew $\mathbf {RSK}$ map of tableaux

In this subsection, we define the skew $\mathbf {RSK}$ map as the result of consecutive operations on pairs of tableaux $P,Q$ . In case $P,Q$ share the same shape, this is equivalent to the definition $\mathbf {RSK}=\iota _2^n$ given in the introduction, as proven in Proposition 3.6 below.

Let P be a semistandard tableau of generalized shape $\lambda / \rho $ . A cell $(c,r) \in \lambda /\rho $ is a corner cell if $(c-1,r),(c,r-1) \notin \lambda /\rho $ . Consider an integer r such that P has a corner cell $(c,r)$ at row r. The internal insertion $\mathcal {R}_{[r]}$ , first introduced in [Reference Sagan and Stanley73], is the operation that constructs tableau $P'=\mathcal {R}_{[r]}(P)$ vacating cell $(c,r)$ of P and inserting value $P(c,r)$ at row $r+1$ following Schensted’s bumping algorithm. For this, we first find $\overline {c} = \min \{k : P(k,r+1)> P(c,r) \}$ . If $\overline {c}$ does not exist, we simply add a $P(c,r)$ -cell at the right of row $r+1$ of P. Alternatively, if $\overline {c}$ exists, we assign cell $(\overline {c},r+1)$ label $P(c,r)$ and we insert, following the same mechanism, $P(\overline {c},r+1)$ at row $r+2$ . Eventually, this algorithm stops, and the result is the tableau $P'$ . It could also happen that we try the internal insertion $\mathcal {R}_{[r]}$ at some row r with only empty cells. In that case, the result is tableaux with an extra empty cell at row r.

The definition of the skew $\mathbf {RSK}$ map of tableaux is given below.

Definition 3.1 (Skew $\mathbf {RSK}$ map of tableaux).

Let $P \in SST(\lambda / \rho ,n)$ , $Q \in SST(\mu / \rho ,m)$ , for some generalized Young diagrams $\lambda ,\mu ,\rho $ . Define the skew $\mathbf {RSK}$ map of $P,Q$

(3.1) $$ \begin{align} \mathbf{RSK}(P,Q) = (P',Q') \in \bigcup_{\varkappa} SST(\varkappa / \mu,n) \times SST(\varkappa / \lambda , m) \end{align} $$

via the following algorithm. Set $P^{(0)} = P$ and $Q^{(0)} = \lambda /\lambda $ , that is, $Q^{(0)}$ has empty shape and external shape equal to $\lambda $ . For $j=1,\dots ,m$ , let $r^{(j)}_1\ge \cdots \ge r^{(j)}_{k_j}$ be the row coordinates of all j-cells of Q and define

(3.2) $$ \begin{align} P^{(j)} = \mathcal{R}_{[r^{(j)}_{k_j}]} \circ \cdots \circ \mathcal{R}_{[r^{(j)}_{1}]} (P^{(j-1)}). \end{align} $$

Then define $Q^{(j)}$ , adding to $Q^{(j-1)} \ j$ -cells so that the external shape of $Q^{(j)}$ matches that of $P^{(j)}$ . Finally, set $P' = P^{(m)}$ and $Q'=Q^{(m)}$ . Sometimes, we will consider the skew $\mathbf {RSK}$ map between standard tableaux $(P,Q)$ , and in such case, we may call this operation skew $\mathbf {RS}$ map.

The reader can check the definition of the skew $\mathbf {RSK}$ map of tableaux with the following example, where for simplicity we have taken $P,Q$ of equal shape

(3.3)

We also report step-by-step calculations

Additional examples are given in Figure 6, right panel, and in Section 1.2.

Next, we present a symmetry of the skew $\mathbf {RSK}$ map of tableaux that was proven in [Reference Sagan and Stanley73] and that will be useful in a number of cases.

Proposition 3.3 says that, like $P'$ and P, also the recording tableau $Q'$ is obtained from Q following a number of internal insertion. This fact is not obvious from Proposition 3.1.

The operation of standardization defined at the end of Section 2.2 is well behaved with respect to the skew $\mathbf {RSK}$ map, as stated in the next proposition. This was already observed in [Reference Sagan and Stanley73] and such natural reduction simplifies proofs of many statements.

Proof. From Proposition 3.1, it is clear that if $(P',Q')=\mathbf {RSK}(P,Q)$ , then

(3.4) $$ \begin{align} \mathbf{RSK}(P, \mathrm{std} (Q)) = (P',\mathrm{std}(Q')). \end{align} $$

Combining this with the symmetry of Proposition 3.3, we conclude the proof.

3.2 Operations $\iota _1,\iota _2$ : internal insertion with cycling

Here, we introduce two operations $\iota _1,\iota _2$ on pairs of tableaux sharing the same shape. They are the same as in the introduction.

Definition 3.5 (Internal insertion with cycling).

Let $(P,Q)$ be a pair of semistandard tableaux with same shape. Let $r_1\ge \cdots \ge r_k$ be the row coordinates of all $1$ -cells of Q. Define

(3.5) $$ \begin{align} \iota_2(P,Q) = (P', Q'), \end{align} $$

where $P' = \mathcal {R}_{[r_k]} \cdots \mathcal {R}_{[r_1]}(P)$ and $Q'$ is obtained from Q vacating first all $1$ -cells, subtracting 1 from the remaining entries and finally adding n-cells so that the final shape equals that of $P'$ . For an example, see equation (1.5). We also define $\iota _1=\mathrm {swap} \circ \iota _2 \circ \mathrm {swap}$ , where $\mathrm {swap}(x,y)=(y,x)$ .

The next proposition states that both $\iota _1$ and $\iota _2$ represent refinements of the skew $\mathbf {RSK}$ map and that the definition of the skew $\mathbf {RSK}$ map of tableaux given in Proposition 3.1 is consistent with the one given in Section 1.2, under the assumption that $P,Q$ have same shape.

Proposition 3.6. For $P,Q\in SST(\lambda / \rho ,n)$ , we have

(3.6) $$ \begin{align} \iota_1^n(P,Q) = \mathbf{RSK}(P,Q) = \iota_2^n (P,Q). \end{align} $$

Proof. We will only prove our claim for pairs of standard tableaux $P,Q$ and for $\epsilon =2$ . This implies the more general case with pairs of semistandard tableaux by Proposition 3.7. On the other hand, the case $\epsilon =1$ follows by the $\epsilon =2$ case since the kernel of a pair of tableaux is invariant under swap, that is, $\ker (P,Q)=\ker (Q,P)$ .

The proof presented below consists in a case-by-case analysis of the insertion procedure, and although rather technical, it is not conceptually involved. We will analyze the internal insertion of a cell in the P tableaux, and we will follow modifications that such insertion imply, showing that, in all cases, the result, paired with the corresponding changes in the Q tableau does not modify the quantity $\ker (P,Q)$ .

For any j, let $p^{(j)}, q^{(j)}, \tilde {p}^{(j)}, \tilde {q}^{(j)}$ be weakly increasing words recording, respectively, the j-th rows of $P, Q, \tilde {P}, \tilde {Q}$ . Assume that the $1$ -cell of Q lies at row r and that the n-cell of $\tilde {Q}$ lies at row $\bar {r}$ , that is, $1\in p^{(r)}$ and $n\in \tilde {p}^{(\bar {r})}$ . Then we have $\tilde {P}=\mathcal {R}_{[r]}(P)$ , and during the internal insertion a new cell gets created at row $\bar {r}$ . Let $\nu =\ker (P,Q)$ and $\tilde {\nu } = \ker (\tilde {P},\tilde {Q})$ . We aim to show that for each $j \ge 1$ we have $\nu _j - \nu _{j+1} = \tilde {\nu }_j - \tilde {\nu }_{j+1}$ . For this, we use the explicit expression of the kernel of a pair of tableaux discussed in Section 2.4, and we have

(3.7) $$ \begin{align} \begin{aligned} \nu_j-\nu_{j+1} &= \rho_j - \lambda_{j+1} + \mathrm{ov}(p^{(j)},p^{(j+1)}) \wedge \mathrm{ov}(q^{(j)},q^{(j+1)}) \\ \tilde{\nu}_j-\tilde{\nu}_{j+1} &= \tilde{\rho}_j - \tilde{\lambda}_{j+1} + \mathrm{ov}(\tilde{p}^{(j)},\tilde{p}^{(j+1)}) \wedge \mathrm{ov}(\tilde{q}^{(j)},\tilde{q}^{(j+1)}), \end{aligned} \end{align} $$

where $\tilde {\lambda } / \tilde {\rho }$ is the the shape of $\tilde {P}, \tilde {Q}$ . Since

(3.8) $$ \begin{align} \tilde{\rho}_j = \rho_j + \delta_{j,r} \qquad \text{and} \qquad \tilde{\lambda}_j = \lambda_j + \delta_{j,\bar{r}}, \end{align} $$

to prove our proposition we need to show that

(3.9) $$ \begin{align} \mathrm{ov}(\tilde{p}^{(j)},\tilde{p}^{(j+1)}) \wedge \mathrm{ov}(\tilde{q}^{(j)},\tilde{q}^{(j+1)} ) = \mathrm{ov}(p^{(j)},p^{(j+1)}) \wedge \mathrm{ov}(q^{(j)},q^{(j+1)}) - \delta_{j,r} + \delta_{j,\bar{r}-1}. \end{align} $$

We start by comparing overlaps between rows of Q and $\tilde {Q}$ . We find that

(3.10) $$ \begin{align} \mathrm{ov}(\tilde{q}^{(j)},\tilde{q}^{(j+1)}) = \mathrm{ov}(q^{(j)},q^{(j+1)}) - \mathbf{1}_{1\in q^{(j)}} + \mathbf{1}_{n \in \tilde{q}^{(j+1)}}, \end{align} $$

which follows from a simple inspection of cycling of letters in the Q tableau, and we only need to take care of rows where a cell is vacated or created. The comparison between $\mathrm {ov}(p^{(j)}, p^{(j+1)})$ and $\mathrm {ov}(\tilde {p}^{(j)}, \tilde {p}^{(j+1)})$ is more laborious, and we need to check for all different choices of j. To simplify our notation, we set $a=p^{(j)}$ , $b=p^{(j+1)}$ , $\tilde {a}=\tilde {p}^{(j)}$ , $\tilde {b}=\tilde {p}^{(j+1)}$ . If $L=\mathrm {ov}(a,b)$ , then $a_i<b_{\ell (b) - L +i}$ for all $i=1,\dots ,L$ and there exists x such that $a_x> b_{\ell (b) - L +x-1}$ . Assume that x is the smallest of such indices, and call $y=\ell (b) - L +x-1$ , as in Figure 4. We call $(x,y)$ a blocking pair of depth L. It is clear that the existence of a blocking pair of depth L is equivalent to saying that $\mathrm {ov}(a,b)=L$ . Notice that such notation also covers extremal cases when $L=\ell (a)$ , and $L=\ell (b)$ where we set, respectively, $x=\ell (a)+1$ and $y=0$ . Let us now confirm equation (3.9) for all cases.

Figure 4 Notation used in the proof of Proposition 3.8. Here, $L=\mathrm {ov}(a,b)$ and $a_x>b_y$ so that x and y form a blocking pair of depth L.

• $j<r-1$ . In this case, $a=\tilde {a}, b=\tilde {b}$ and by equations (3.10), (3.9) holds.

• $j=r-1$ . Here, we have $\tilde {a}=a$ and $\tilde {b}=b_2 \cdots b_{\ell (b)}$ . We set $\tilde {x}=x$ and $\tilde {y} = y-1$ , and this is a blocking pair of depth L for $\tilde {a},\tilde {b}$ , whenever $y \neq 0$ . If on the other hand $y=0$ , we have $L=\ell (b)$ and since $1 \in q^{(r)}$ , then $L>\mathrm {ov}(q^{(r-1)},q^{(r)}) = \mathrm {ov}(\tilde {q}^{(r-1)},\tilde {q}^{(r)})$ , by equation (3.10). In both cases, equation (3.9) holds.

• $j=r$ . In this case, $\tilde {a}=a_2 \dots a_{\ell (a)}$ and $\tilde {b}=b_1 \cdots b_{\bar {k}} \, a_1 \, b_{\bar {k}+2} \cdots b_{\ell (b)}$ for an index $\bar {k} \in \{0,\dots ,\ell (b)\}$ . If $\bar {k}=\ell (b)$ , then $\mathrm {ov}(\tilde {a},\tilde {b})=L=0$ and by equation (3.10), equation (3.9) holds. Assume now that $\bar {k}<\ell (b)$ . If $x=1$ , then necessarily $\bar {k} = y$ and $\tilde {x}=1$ , $\tilde {y}=y+1$ is a blocking pair of depth $L-1$ for $\tilde {a},\tilde {b}$ . If on the other hand $x\neq 1$ , then $\bar {k} \le y$ , and in such case we set $\tilde {x}=x-1, \tilde {y}=y$ , which again is a blocking pair of depth $L-1$ for $\tilde {a},\tilde {b}$ . Overall, we have shown that $\mathrm {ov}(\tilde {a},\tilde {b}) = \mathrm {ov}(a,b) -1 +\delta _{\bar {k} , \ell (b)}$ , which confirms equation (3.9).

• $r<j<\bar {r}-1$ . We have $\tilde {a}=a_1 \cdots a_{\bar {m}-1} \, z \, a_{\bar {m}+1} \cdots a_{\ell (a)}$ for some letter z and some index $\bar {m}$ . Similarly, we have $\tilde {b}=b_1 \cdots b_{\bar {k}} \, a_{\bar {m}} \, b_{\bar {k}+2} \cdots b_{\ell (b)}$ for an index $\bar {k} \in \{0,\dots ,\ell (b)\}$ . If $\bar {m}>x$ , then $\bar {k} \ge y$ and we set $\tilde {x}=x,\tilde {y}=y$ . If $\bar {m}=x$ , then $\bar {k}=y$ and we set $\tilde {x}=x+1,\tilde {y}=y+1$ . Lastly, if $\bar {m}<x$ necessarily $\tilde {b}_y < \tilde {a}_x$ and we set $\tilde {x}=x,\tilde {y}=y$ . In all cases, $\tilde {x},\tilde {y}$ form a blocking pair of depth L for $\tilde {a},\tilde {b}$ , confirming equation (3.9).

• $r<j=\bar {r}-1$ . Observe that the case $r=j=\bar {r}-1$ was already treated above. We have $\tilde {a}=a_1 \cdots a_{\bar {m}-1} \, z \, a_{\bar {m}+1} \cdots a_{\ell (a)}$ for some letter z and some index $\bar {m}$ and $\tilde {b}=b_1\cdots b_{\ell (b)} \, a_{\bar {m}}$ . Since $a_{\bar {m}}>b_k$ for all k, we necessarily have $\bar {m}>L$ and $\tilde {x}=x,\tilde {y}=y$ becomes a blocking pair of depth $L+1$ for $\tilde {a},\tilde {b}$ . Hence, from equation (3.10), (3.9) holds.

• $j=\bar {r}$ . In this case, $\tilde {a}=a_1\cdots a_{\ell (a)} z$ for some letter z greater than all entries of a and $\tilde {b}=b$ . If $\ell (a)<L$ then necessarily $\mathrm {ov}(\tilde {a},\tilde {b}) = \mathrm {ov}(a,b)$ , which implies equation (3.10). If on the other hand $\ell (a) = L$ , then it could happen that $\mathrm {ov}(\tilde {a},\tilde {b}) =L+1$ , if entries of a are small and $z<b_{\ell (b)}$ . Nevertheless, since

  (3.11) $$ \begin{align} \ell(a) = \ell(q^{(\bar{r})}) \ge \mathrm{ov}(q^{(\bar{r})}, q^{(\bar{r}+1)}) = \mathrm{ov}( \tilde{q}^{(\bar{r})}, \tilde{q}^{(\bar{r}+1)}), \end{align} $$
  we always have $\mathrm {ov} (\tilde {q}^{(\bar {r})}, \tilde {q}^{(\bar {r}+1)}) \le L$ and equation (3.9) holds.

• $j>\bar {r}$ . Here, $\tilde {a}=a$ and $\tilde {b}= b$ so that equation (3.9) trivially holds.

The previous list of checks exhausts all the cases and completes the proof.

3.3 The skew RS map of arrays

In this subsection, we introduce the skew $\mathbf {RS}$ map of pairs of arrays, following [Reference Viennot89]. Due to the correspondence between arrays and standard tableaux, it provides a diagrammatic realization of the skew $\mathbf {RS}$ map for standard tableaux, as will be seen in Proposition 3.14. It represents a reformulation of the shadow line construction by Viennot [Reference Viennot88], as we see now.

For an example of a shadow line construction, see Figure 5. There in correspondence to intersection points of lines with the same color C we drew bullets of color $C+1$ . The rules of assigning colors to lines in this construction can be translated into local rules of configurations on edges of the lattice [Reference Fomin27, Reference Viennot89].

Figure 5 The shadow line construction for and , equivalent to the skew $\mathbf {RS}$ map in the left panel of Figure 6

Figure 6 In the left panel, we see the graphical representation of the skew $\mathbf {RS}$ map between , . We have colored each edge of the grid based on its value. Black bullets denote faces where north and east edges take simultaneously value 1 defining partial permutation $\pi^{(1)}=\left( \begin{smallmatrix} 2 & 4 \\ 4 & 1 \end{smallmatrix} \right) $ . In the right panel, we reported on the left and bottom sides the tableaux $(P,Q)$ and on the right and top sides tableaux $(P',Q') = \mathbf {RS}(P,Q)$ . One can check that and .

Definition 3.10 ( $\mathbb {Z}$ -valued edge configurations).

On a planar lattice $\Lambda \subseteq \mathbb {Z}\times \mathbb {Z}$ , $\mathbb {Z}$ -valued edge configurations $\mathcal {E}$ are quadruples of functions $(\mathsf {W},\mathsf {S},\mathsf {E},\mathsf {N}):\Lambda \to \mathbb {Z}$ such that $\mathsf {E}(c) = \mathsf {W}(c+\mathbf {e}_1)$ and $\mathsf {N}(c) = \mathsf {S}(c+\mathbf {e}_2)$ for all points $c\in \Lambda $ where these conditions make sense. We say that $\mathcal {E}$ is admissible if it satisfies the local rules

(3.12) $$ \begin{align} &1.\quad \mathsf{E}(c) = \mathsf{W}(c) \text{ and } \mathsf{N}(c) = \mathsf{S}(c), \text{ if } \mathsf{W}(c) \ne \mathsf{S}(c);\\&2.\quad \mathsf{E}(c) = \mathsf{N}(c) = \mathsf{S}(c)+1, \text{ if } \mathsf{S}(c)=\mathsf{W}(c).\nonumber \end{align} $$

In this language, we define the skew $\mathbf {RS}$ map of arrays as follows.

For an example of edge configurations and an evaluation of the skew $\mathbf {RS}$ map of two arrays, corresponding to Figure 5, see Figure 6, left panel. Note that positions $(j,i)$ of bullets of color k are determined by the condition $\mathsf {N}(j,i) = \mathsf {E}(j,i) =k$ . We will encode the positions $(j_1,i_1),(j_2,i_2),\dots $ of bullets of color k in the partial permutation

(3.13) $$ \begin{align} \pi^{(k)} = \left( \begin{matrix} j_1 & j_2 & \cdots \\ i_1 & i_2 & \cdots \end{matrix} \right). \end{align} $$

Using edge configurations, we also define operators of $\iota _1,\iota _2$ on pairs of arrays.

Definition 3.12 ( $\iota _1,\iota _2$ on arrays).

Let be a pair of arrays that are permutations of each other, and consider on $\Lambda _{n,n}$ the unique admissible edge configuration with for all $i=1,\dots ,n$ . Define setting

(3.14)

Analogously, define .

Comparing 3.11 and 3.12, we see that $\mathbf {RS}=\iota _1^n=\iota _2^n$ holds as operations on pairs of arrays.

The next proposition shows that $\iota _1,\iota _2$ , both as operations on pairs of standard tableaux and on pairs of arrays, coincide, modulo row-coordinate parameterization.

Proof. We prove our statement only for the case $\epsilon =2$ , as the remaining case is analogous. The fact that $\nu =\ker (P,Q)$ does not change after the application of $\iota _2$ was shown in Proposition 3.8. Therefore, we need to show that if $(\tilde {P},\tilde {Q})=\iota _2(P,Q)$ and , we have .

Call

, so that $\tilde {P}=\mathcal {R}_{[r]}(P)$ . Suppose that during the internal insertion bumping happens k times and cells of P at location $(c_j,r+j)$ move to $(c_{j+1},r+j+1)$ for $j=0,\dots ,k-1$ . Denoting $p_j=P(c_j,r+j)$ , we have $c_0 \ge c_1 \ge \cdots \ge c_k$ and $p_0 < p_1 < \cdots < p_{k-1}$ . In the row-coordinate array

, this implies that

and importantly

for $\ell =1,\dots , p_{j+1} - p_j$ . We also have

(3.15)

On the other hand, we now draw the edge configuration on $\Lambda _{n,n}$ corresponding to the pair

as for equation (3.12) and we see that $S(1,i)=W(1,i)=r+j,p_j\leq i<p_{j+1},0\leq j\leq k-1$ and $E(1,i)=N(1,i)=r+j+1$ . Comparing this with the definition of $\iota _2$ in equation (3.14), we can confirm that

holds.

To show that notice that, from the cycling of labels, the i-cell of $\tilde {Q}$ lies at row for $i=1,\dots ,n-1$ . On the other hand, the n-cell of $\tilde {Q}$ lies at row $r+k$ , which, by the computation above is also the value of . This completes the proof.

The coincidence extends to the skew $\mathbf {RS}$ maps.

We finally report a simple ‘restriction property’ of the shadow line construction. For the next proposition, we need the notion of partial arrays, which are elements of $(\mathbb {Z} \cup \{\varnothing \})^n$ . Given an array , we denote by the partial array with entries if and otherwise.

3.4 The skew RSK map of matrices

We extend the operation of skew $\mathbf {RS}$ map to include infinite matrices of integers. Thanks to the correspondence between matrices and tableaux of Proposition 2.3, this provides a diagrammatic realization of the skew $\mathbf {RSK}$ map for tableaux. By the standardization most of their properties follow directly from the ones of the skew $\mathbf {RS}$ map.

We can generalize the shadow line construction of Proposition 3.9 by allowing edges of the lattice $\Lambda _{m,n}$ to be crossed by arbitrary many lines. We impose that on each cell for any two lines crossing the same horizontal (resp. vertical) edge the one with higher color stays at the left (resp. below) of the one with lower color; see Figure 7. This implies also that for each color bullets at a cell lie strictly to the right of those at cells $c-k \mathbf {e}_2$ and strictly above those at cells $c-k \mathbf {e}_1$ for all $k\in \mathbb {N}_0$ . We record the list of colored lines crossing a specific edge in an array $v\in \mathbb {V}$ and for any $C\in \mathbb {Z}$ the value $v_C$ will count the number of C-colored lines; see Figure 8. Such discussion justifies the following definition.

Figure 7 Generalized shadow line construction on the lattice $\Lambda _{3,3}$ .

Figure 8 On the left panel, we see the equivalence between $\mathbb {V}$ -valued edge configurations and configurations of colored lines through a face. On the right panel, a graphical interpretation of local rules (3.16)

Definition 3.18 ( $\mathbb {V}$ -valued edge configurations).

For a planar lattice $\Lambda \subseteq \mathbb {Z}\times \mathbb {Z}$ , define $\mathbb {V}$ -valued edge configurations $\mathcal {E}$ as quadruples of functions $(\mathsf {W},\mathsf {S},\mathsf {E},\mathsf {N}):\Lambda \to \mathbb {V}$ such that $\mathsf {E}(c)=\mathsf {W}(c+\mathbf {e}_1)$ and $\mathsf {N}(c)=\mathsf {S}(c+\mathbf {e}_2)$ for all $c\in \Lambda $ . An edge configuration is admissible if it satisfies the local rules

(3.16) $$ \begin{align} \begin{aligned}&1. \ \mathsf{E}_j(c) = \mathsf{W}_j(c) - \mathsf{S}_j(c) \wedge \mathsf{W}_j(c) + \mathsf{S}_{j-1}(c) \wedge \mathsf{W}_{j-1}(c), \\&2. \ \mathsf{N}_j(c) = \mathsf{S}_j(c) - \mathsf{S}_j(c) \wedge \mathsf{W}_j(c) + \mathsf{S}_{j-1}(c) \wedge \mathsf{W}_{j-1}(c),\end{aligned} \end{align} $$

for all $c\in \Lambda $ , $j\in \mathbb {Z}$ . Here $a\wedge b=\min (a,b)$ .

Local rules (3.16) describe the arrangement of lines of the each color j around single faces of the lattice. Namely, $\mathsf {W}_j(c)$ is the number of j-colored lines entering face c from the left and similarly for $\mathsf {S}_j, \mathsf {E}_j, \mathsf {N}_j$ . We report this statement in the next proposition, and we refer to Figure 8 for the graphical interpretation of such rules.

In line with Section 2.6, we can define the standardization of an admissible $\mathbb {V}$ -valued edge configuration $\mathcal {E}$ on a lattice $\Lambda $ . This will be the admissible $\mathbb {Z}$ -valued edge configuration $\mathcal {E}'=\mathrm {std}(\mathcal {E})$ on the lattice $\Lambda '$ obtained from $\mathcal {E}$ ‘blowing up’ faces of $\Lambda $ as prescribed by Proposition 3.19.

Definition 3.20 (Skew $\mathbf {RSK}$ map of matrices).

Let $\alpha \in \mathbb {M}_{n\times \infty },\beta \in \mathbb {M}_{m\times \infty }$ , and consider the unique $\mathbb {V}$ -valued admissible edge configuration on $\Lambda _{m,n}$ such that $\mathsf {W}_k(1,i)=\alpha _{i,k}$ , $\mathsf {S}_k(j,1)=\beta _{j,k}$ , for all $i=1,\dots ,n$ , $j=1,\dots ,m$ , $k\in \mathbb {Z}$ . We define the skew $\mathbf {RSK}$ map

$$ \begin{align*} \mathbf{RSK}(\alpha,\beta)=(\alpha',\beta') \in \mathbb{M}_{n\times \infty} \times \mathbb{M}_{m \times \infty}, \end{align*} $$

as the pair of matrices $\alpha _{i,k}^{\prime }=\mathsf {E}_k(m,i)$ , $\beta _{j,k}^{\prime }=\mathsf {N}_k(j,n)$ , for all $i=1,\dots ,n$ , $j=1,\dots ,m$ , $k\in \mathbb {Z}$ . From the configuration, define also the family of matrices $M^{(k)}$ as

(3.17) $$ \begin{align} M^{(k)}(j,i) = \mathsf{N}_k(j,i) \wedge \mathsf{E}_k(j,i). \end{align} $$

Example 3.21. Define matrices

(3.18)

We evaluate $(\alpha ',\beta ') = \mathbf {RSK}(\alpha , \beta )$ computing the corresponding $\mathbb {V}$ -valued edge configuration on $\Lambda _{3,3}$ . The result is reported in Figure 7, and we have

(3.19)

To configuration of Figure 7, we associate matrices $M^{(k)}$ as described by equation (3.17). For instance, from the same figure one can check that $M^{(1)}=\left ( \begin {smallmatrix} 0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & 1 & 1 \end {smallmatrix} \right )$ or $M^{(2)}=\left ( \begin {smallmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {smallmatrix} \right )$ .

Definition 3.22. For a pair of matrices $(\alpha ,\beta ) \in \mathcal {M}_n$ , construct on $\Lambda _{n,n}$ the corresponding $\mathbb {V}$ -valued edge configuration as in Proposition 3.20. We define $\iota _2(\alpha ,\beta ) = (\tilde {\alpha }, \tilde {\beta })$ , setting for all $k\in \mathbb {Z}$

(3.20) $$ \begin{align} \tilde{\alpha}_{i,k} = \mathsf{W}_k(2,i) \qquad \text{for } i=1\dots,n \qquad \text{and} \qquad \tilde{\beta}_{i,k} = \begin{cases} \mathsf{S}_k(i+1,1) \quad &\text{if } i=1,\dots,n-1, \\ \mathsf{N}_k(1,n) \quad &\text{if } i=n. \end{cases} \end{align} $$

Operator $\iota _1$ is defined by duality $\iota _1(\alpha ,\beta )=\mathrm {swap} \circ \iota _2 \circ \mathrm {swap} (\alpha ,\beta )$ .

Notice that, in the previous definition, also the pair $(\tilde {\alpha },\tilde {\beta })$ belongs to the set $\mathcal {M}_n$ .

Geometric interpretation of operators $\iota _1,\iota _2$ provided by Proposition 3.23 yields a visual proof of the nontrivial fact that they commute with each other.

Proof. We first prove that $\iota _1$ and $\iota _2$ commute when they act on pairs of matrices. Consider the admissible $\mathbb {V}$ -valued edge configuration on the lattice $\Lambda _{n+1,n+1}$ such that for all $k\in \mathbb {Z}$

$$ \begin{align*} \mathsf{W}_k(1,i) = \alpha_{i,k}, \qquad \mathsf{S}_k(1,i) = \beta_{i,k}, \qquad \text{for } i=1,\dots,n \\ \text{and} \qquad \mathsf{W}_k(1,n+1) = \mathsf{E}_k(n,1) \qquad \mathsf{S}_k(n+1,1) = \mathsf{N}_k(1,n). \end{align*} $$

In this lattice, $\iota _1$ and $\iota _2$ act, respectively, as upward and rightward shift so that defining $\tilde {\alpha }$ , $\tilde {\beta }$ as

(3.21) $$ \begin{align} \tilde{\alpha}_{i,k} = \mathsf{W}_k(2,i+1), \qquad \tilde{\beta}_{i,k} = \mathsf{N}_k(i+1,2), \qquad \text{for } i=1,\dots,n \text{ and } k\in \mathbb{Z} \end{align} $$

we easily see that $(\tilde {\alpha } , \tilde {\beta }) = \iota _1 \circ \iota _2 (\alpha ,\beta ) = \iota _2 \circ \iota _1 (\alpha ,\beta )$ . This proves the proposition for the case of action on pairs of matrices. By Proposition 3.23, the same commutation holds when $\iota _1,\iota _2$ act on pair of semistandard tableaux.

The next proposition gives a restriction property analogous to Proposition 3.16. For any matrix $\alpha \in \mathbb {M}_{n,\infty }$ , we define its truncation $\delta _{\ge k}(\alpha ) = ( \mathbf {1}_{j\ge k} \alpha _{i,j} :i=1,\dots ,n,j\in \mathbb {Z})$ .

4.1 The skew $\mathbf {RSK}$ dynamics

The following definition was sketched in the introductory chapter.

Definition 4.1 (Skew $\mathbf {RSK}$ dynamics).

We define a deterministic dynamics on the space of pairs of generalized tableaux by iterating the skew $\mathbf {RSK}$ map. Fix $P,Q\in SST(\lambda / \rho ,n)$ , and define the skew $\mathbf {RSK}$ dynamics with initial data $(P,Q)$ as

(4.1) $$ \begin{align} \begin{cases} (P_{t+1},Q_{t+1})=\mathbf{RSK}(P_t,Q_t), \qquad t \in \mathbb{Z}, \\ (P_1,Q_1)=(P,Q). \end{cases} \end{align} $$

Analogously, we define the skew $\mathbf {RSK}$ dynamics on the space of pairs of matrices. For a fixed initial state $(\alpha ,\beta )\in \mathcal {M}_n$ , we have

(4.2) $$ \begin{align} \begin{cases} (\alpha^{(t+1)},\beta^{(t+1)})=\mathbf{RSK}(\alpha^{(t)},\beta^{(t)}), \qquad t \in \mathbb{Z}, \\ (\alpha^{(1)},\beta^{(1)})=(\alpha,\beta). \end{cases} \end{align} $$

Since the skew $\mathbf {RSK}$ map is invertible, we see that each realization of the dynamics is uniquely characterized by its initial state. This observation is very powerful as it implies that specific properties of tableaux $(P,Q)$ can be deduced observing their state at an arbitrary time of the skew $\mathbf {RSK}$ dynamics. As already pointed out in Section 1.2, dynamics (4.1) presents conservation laws, that we will characterize in Section 6 and Section 9 below. Moreover, in Section 7 we will devise a linearization technique of the dynamics, which one can regard as a combinatorial variant of the inverse scattering method for classical integrable systems. The study of asymptotic states of the dynamics will be exceptionally revealing.

4.2 Edge configurations on the twisted cylinder

In the Section 3, we have seen the equivalence between the two versions of the skew $\mathbf {RSK}$ maps on tableaux and on matrices through edge local rules on a finite rectangular lattice. Here, we introduce an infinite lattice with certain periodicity and edge configurations compatible with it. This will lead us to define a dynamics on the same lattice, which is closely related to the skew $\mathbf {RSK}$ dynamics.

Definition 4.2 (Twisted cylinder).

The twisted cylinder is the periodic lattice $\mathscr {C}_n= \mathbb {Z}^2 / \sim _n$ , where $(j,i)\sim _n(j', i')$ if $(j',i') = (j+kn,j-kn)$ for some $k\in \mathbb {Z}$ . Natural representations of $\mathscr {C}_n$ we will use are (see Figure 9):

(4.3) $$ \begin{align} \begin{aligned}&\kern-11pt\text{the infinite vertical strip } \{ 1, \dots , n\} \times \mathbb{Z}, \text{ where we impose faces } (n,i) \text{ and } (1,i+n) \text{ to be }\\&\kern-11pt\text{adjacent for all } j.\end{aligned} \end{align} $$

(4.4) $$ \begin{align} \begin{aligned}&\text{the infinite horizontal strip } \mathbb{Z} \times \{ 1, \dots , n\} \text{ where we impose faces } (j,n) \text{ and } (j+n,1) \text{ to be }\\&\text{adjacent for all } j.\end{aligned} \end{align} $$

Figure 9 Two graphical representation of the twisted cylinder $\mathscr {C}_3$ . Blue line represents an up-right path, while red line a down-right loop.

A down-right loop $\xi $ on $\mathscr {C}_n$ is a sequence $(\xi _k: k\in \{1,\dots ,2n\}) \subset \mathscr {C}_n$ , such that

(4.5) $$ \begin{align} \xi_{k+1} \sim_n \xi_{k} + \mathbf{e}_1, \qquad \text{or} \qquad \xi_{k+1} \sim_n \xi_{k} - \mathbf{e}_2, \end{align} $$

for $k=1,\dots , 2n$ , where indices are taken $\mod 2n$ . An example of a down-right loop is the red path drawn in Figure 9, having the form

$$ \begin{align*}c \to c+\mathbf{e}_1 \to \cdots \to c+n\mathbf{e}_1 \to c+n\mathbf{e}_1-\mathbf{e}_2 \to \cdots \to c+n\mathbf{e}_1-(n-1)\mathbf{e}_2 \to c \end{align*} $$

for some $c\in \mathscr {C}_n$ . Assigning edge values along a down-right loop $\xi $ automatically determines the full configuration $\mathcal {E}$ on $\mathscr {C}_n$ as a result of local rules (3.16). We use this to visualize the skew $\mathbf {RSK}$ dynamics on the set of matrices as an edge configuration on $\mathscr {C}_n$ . To any pair $(\alpha , \beta ) \in \mathcal {M}_n$ we associate the edge configuration $\mathcal {E}$ on $\mathscr {C}_n$ identified by the assignment

(4.6) $$ \begin{align} (\alpha , \beta) \mapsto \mathcal{E} \qquad \colon \qquad \mathsf{E}_k(n, i) = \alpha_{i,k}, \qquad \mathsf{N}_k (i,n) = \beta_{i,k}, \qquad \text{for } i=1,\dots,n, \,\,k\in \mathbb{Z}. \end{align} $$

The subclass of admissible edge configurations accessible through mapping (4.6) is defined next.

Proposition 4.5. Let $(\alpha ^{(t)},\beta ^{(t)})$ be the skew $\mathbf {RSK}$ dynamics with initial data $(\alpha ^{(1)},\beta ^{(1)}) =(\alpha ,\beta ) \in \mathcal {M}_n$ , and let $\mathcal {E}$ be the configuration associated to $(\alpha ,\beta )$ by equation (4.6). Then, for all $i=1,\dots ,n$ and $t \in \mathbb {Z}$ we have $(\alpha ^{(t)},\beta ^{(t)}) = (\alpha ,\beta )_{(1,tn+1)}$ ; see Figure 10.

Figure 10 A visualization of the skew $\mathbf {RSK}$ dynamics of matrices $(\alpha ^{(t)},\beta ^{(t)})$ as edges of a configuration $\mathcal {E}$ . Faces of $\mathscr {C}_n$ have coordinates $(j,i-kn)$ .

Figure 11 In the left panel, we see the evaluation of Viennot map transforming $\overline {\pi }$ of equation (4.9). This is done through the periodic shadow line construction explained in Proposition 4.6 and represented in the right panel. We made cells of $\mathscr {C}_n$ fatter in order to allocate multiple bullets while letting them keep the correct relative positions.

4.3 Periodic shadow line construction and Viennot dynamics

Edge configurations $\mathcal {E} \in \mathfrak {E}_n$ , in analogy with the finite case, identify families of compactly supported maps $\overline {M}^{(t)}:\mathscr {C}_n \to \mathbb {N}_0$ assigning

(4.7) $$ \begin{align} \overline{M}^{(t)}(c) = \mathsf{N}_t(c) \wedge \mathsf{E}_t(c). \end{align} $$

On the other hand, as proven in Proposition 4.9 below, for any map $\overline {M} \in \overline {\mathbb {M}}_{n\times n}$ we can construct the family $\overline {M}^{(t)}$ , with the convention that $\overline {M}^{(1)}=\overline {M}$ and the configuration $\mathcal {E}$ such that equation (4.7) holds. Such procedure constitutes a periodic variant of the Viennot shadow line construction [Reference Viennot88] or of Fulton’s matrix ball construction [Reference Fulton31].

Definition 4.6 (Viennot map).

Let $\overline {M}\in \overline {\mathbb {M}}_{n\times n}$ . At each face $c \in \mathscr {C}_n$ , allocate $\overline {M}(c)$ black bullets and apply the shadow line construction explained at the beginning of Section 3.4, letting each bullet emanate two black rays in the north and east directions. By periodicity of $\mathscr {C}_n$ and the fact that there are only a finite number of black bullets, each ray terminates somewhere intersecting with another. The collection of such mutual intersections of rays determines a new generation of red bullets (see Figure 11, right panel), and we define

(4.8) $$ \begin{align} \overline{M}'(c) = \# \text{ of new generation bullets at cell } c. \end{align} $$

The map $\mathbf {V} :\overline {M} \mapsto \overline {M}'$ takes the name of Viennot map. Using correspondence (2.6), we define the action of Viennot map also on weighted biwords $\mathbf {V}(\overline {\pi }) = \overline {\pi }'$ , imposing $\overline {M}(\overline {\pi }')= \mathbf {V}(\overline {M}(\overline {\pi }))$ .

An example of evaluation of map $\mathbf {V}$ is reported in Figure 11(a). There, we see the transition

(4.9) $$ \begin{align} \overline{\pi} = \left( \begin{matrix} 1 & 1 & 2 & 2 & 2 & 2 & 3 \\ 1 & 2 & 1 & 1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \end{matrix} \right) \xrightarrow[]{\hspace{.6cm}} \mathbf{V}(\overline{\pi}) = \left( \begin{matrix} 1 & 1 & 2 & 2 & 2 & 2 & 3 \\ 1 & 3 & 1 & 1 & 1 & 2 & 1 \\ 0 & 0 & 0 & -1 & -1 & 1 & 0 \end{matrix} \right). \end{align} $$

The shadow lines produced by the computation of $\overline {\pi } \mapsto \mathbf {V}(\overline {\pi })$ consist in a sequence of connected broken lines as we see in Figure 11(b). Recording the cells of $\mathscr {C}_n$ visited by each of these broken lines, we naturally define a sequence of down-right loops $\xi ^{(1)},\xi ^{(2)},\dots $ . We will use these loops later in Section 6. For instance, in Figure 11 (b), the two topmost loops have the same form

(4.10) $$ \begin{align} (1,4) \to (2,4) \to (2,3) \to (2,2) \to (2,1) \to (3,1) \to (1,4). \end{align} $$

The map $\mathbf {V}$ can be thought as a generator of a deterministic dynamics on the space $\overline {\mathbb {M}}_{n\times n}$ .

Definition 4.7 (Viennot dynamics).

Fix $\overline {M} \in \overline {\mathbb {M}}_{n\times n}$ , and define the Viennot dynamics with initial data $\overline {M}$ as

(4.11) $$ \begin{align} \begin{cases} \overline{M}^{(t+1)} = \mathbf{V}(\overline{M}^{(t)}),\qquad t\in \mathbb{Z} \\ \overline{M}^{(1)} = \overline{M}. \end{cases} \end{align} $$

Analogously, for $\overline {\pi }\in \overline {\mathbb {A}}_{n,n}$ , one defines the Viennot dynamics $\overline {\pi }^{(t+1)}=\mathbf {V}(\overline {\pi }^{(t)})$ with initial data $\overline {\pi }^{(1)}=\overline {\pi }$ .

Proof. The bijection between $\mathfrak {E}_n$ and $\mathcal {M}_n$ is proven in Proposition 4.4. We then need to prove that the assignment

(4.12) $$ \begin{align} \overline{M}(c) = \mathsf{N}_1(c) \wedge \mathsf{E}_1(c) \end{align} $$

defines a bijection $\overline {M} \mapsto \mathcal {E}$ between $\overline {\mathbb {M}}_{n \times n}$ and $\mathfrak {E}_n$ . Let $\overline {M}^{(t)}$ be the Viennot dynamics with initial data $\overline {M}$ . Then for any $c\in \mathscr {C}_n$ we set $\mathsf {W}_t(c)$ as the number of lines of the construction $\overline {M}^{(t)} \to \overline {M}^{(t+1)}$ entering the face c from west and similarly for $\mathsf {S}_t,\mathsf {E}_t,\mathsf {N}_t$ . From such edge configuration, define $(\alpha ^{(t)},\beta ^{(t)}) = (\alpha ,\beta )_{(1,tn+1)}$ , following the notation of Proposition 4.3 and recall the truncation operator $\delta _{\ge 1}$ from Proposition 3.26. Since $\overline {M}$ is compactly supported there exists $k^*> 0$ such that $\overline {M}_{i,j}(k)=0$ for all $i,j\in \{1,\dots ,n\}$ and $|k|>k^*$ . This implies that $\delta _{\ge 1}(\alpha ^{(-k^*)}) = \delta _{\ge 1}(\beta ^{(-k^*)}) = 0$ and $\delta _{\ge 1}(\alpha ^{(k^*)}) = \alpha ^{(k^*)}$ , $\delta _{\ge 1}(\beta ^{(k^*)}) = \beta ^{(k*)}$ . By recursive application of Proposition 3.26, we find that $(\alpha ^{(k^*)},\beta ^{(k^*)}) \in \mathcal {M}_n$ and they are uniquely determined. This implies that $\mathcal {E}\in \mathfrak {E}_n$ completing the proof. Now that we have constructed the correspondence $\overline {M} \leftrightarrow \mathcal {E}$ , we can associate to the matrix $\overline {M}$ the pair $(\alpha ,\beta )$ as in equation (4.6).

If a pair $(\alpha ,\beta ) \in \mathcal {M}_n$ and a matrix $\overline {M} \in \overline {\mathbb {M}}_{n \times n}$ are in correspondence through the bijection described in Proposition 4.9, we will use the notation . Moreover, composing such correspondence with the row-coordinate parameterization $(P,Q) \xrightarrow []{} (\alpha ,\beta )$ defines a projection denoted by

(4.13) $$ \begin{align} (P,Q) \xrightarrow[]{\mathbf{SS} \,} \overline{M}. \end{align} $$

An analogous projection $(P,Q) \xrightarrow []{\mathbf {SS} \,} \overline {\pi } \in \overline {\mathbb {A}}_{n,n}$ is defined taking advantage of mapping (2.6).

By the same arguments as in proof of Proposition 4.9, $\mathcal {M}_n^+$ and $\overline {\mathbb {M}}_{n \times n}^+$ are also in bijection. Combining this with the bijection between $\mathcal {M}_n^+\times \mathbb {Y}$ and pairs $(P,Q)$ of classical semistandard tableaux reported in Proposition 2.3 gives the Sagan–Stanley correspondence stated below.

Theorem 4.11 ([Reference Sagan and Stanley73], Theorem 6.6).

There exists a canonical bijection

(4.14)

which we denote by . Moreover, the property

(4.15) $$ \begin{align} |\rho| = \mathrm{wt}(\overline{M}) + |\nu| \end{align} $$

holds.

4.4 Relations between skew $\mathbf {RSK}$ and Viennot dynamics

The Viennot dynamics enjoys a very simple relations with the skew $\mathbf {RSK}$ dynamics which we describe in the two propositions below.