# A symmetry property for \(q\)-weighted Robinson–Schensted and other branching insertion algorithms

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## Abstract

In [19], a \(q\)-weighted version of the Robinson–Schensted algorithm was introduced. In this paper, we show that this algorithm has a symmetry property analogous to the well-known symmetry property of the usual Robinson–Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin [5, 6, 7, 8]. This approach, which uses ‘growth graphs’, can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other \(q\)-weighted versions of the Robinson–Schensted algorithm which have recently been introduced by Borodin–Petrov [2].

## Keywords

Robinson–Schensted algorithm Growth diagram Q-analogue Permutation## Mathematics Subject Classification

05C85 05E10 05E05## 1 Introduction

In [19], a \(q\)-weighted version of the Robinson–Schensted (RS) algorithm was introduced. In this paper, we show that this algorithm enjoys a symmetry property analogous to the well-known symmetry property of the RS algorithm. The proof uses a generalisation of the growth diagram approach introduced by [5, 6, 7, 8].

The insertion algorithm we consider in this paper is based on column insertion, but the technique applies to any insertion algorithm belonging to a certain class of ‘branching insertion algorithms’, as described in Sect. 7 below. For example, [2] have recently introduced a \(q\)-weighted version of the row insertion algorithm, which is defined similarly to the column insertion version of [19]; they also consider a wider family of such algorithms (and, more generally, dynamics on Gelfand–Tsetlin patterns), some of which fall into the framework considered in the present paper, and can similarly be shown to have the symmetry property. We discuss such extensions in Sect. 7 below.

The Robinson–Schensted algorithm is a combinatorial algorithm which was introduced by Robinson [1] and Schensted [22]. It has wide applications in representation theory and probability theory, e.g. last passage percolation, totally asymmetric simple exclusion process/corner growth model, random matrix theory [12], queues in tandem [17] and more.

There are two versions of RS algorithms, the row insertion and column insertion version. In most of this paper, we deal with column insertion and its \(q\)-version. The RS algorithm transforms a word to a tableau pair of the same shape. A word can be treated as a path, and hence a random word corresponds to a random walk. When taking such a random walk, the shape of the output tableaux is a Markov chain, whose transition kernel is related to the Schur symmetric functions [16]. A geometric generalisation transforms a Brownian motion with drift to a Markov process whose generator is related to \(\mathsf {GL}(n,\mathbb R)\)-Whittaker functions [18], which are eigen functions of the quantum Toda chain [14]. The \(q\)-Whittaker functions on the one hand are a generalisation of the Schur symmetric functions and a specialisation of Macdonald \((q,t)\)-symmetric functions when \(t=0\) [15], and on the other hand are eigen functions of the \(q\)-deformed quantum Toda chain [4, 20]. When \(q\rightarrow 0\) they become the Schur functions, and when \(q\rightarrow 1\) with a proper scaling [10] they converge to Whittaker functions. In the spirit of this connection, a \(q\)-weighted RS algorithm was formulated in [19], which transforms the random walk to a Markov chain that is related to \(q\)-Whittaker functions.

As is expected, the algorithm degenerates to the usual RS algorithm when \(q\rightarrow 0\). Part of the random \(P\)-tableau also has \(q\)-TASEP dynamics [19], the latter introduced in [24], just as the same part of the random \(P\)-tableau of usual RS algorithm has TASEP dynamics [17].

Recently, a \(q\)-version of the RS row insertion algorithm was also introduced in [2]. It denegerates to the usual RS algorithm with row insertion when \(q\rightarrow 0\).

In this paper, we show that both algorithms enjoy a symmetry property analogous to the well-known symmetry property of the RS algorithm. Basically, the symmetry property for the RS algorithms restricted to permutation inputs is the property that the output tableau pair is interchanged if the permutation is inversed. Knuth [13] generalised the usual row insertion algorithm to one which takes matrix input, which we refer to as Robinson–Schensted–Knuth (RSK) algorithm. For this algorithm, the symmetry property is that the output tableau pair is interchanged if the matrix is transposed. Note that the matrix becomes the permutation matrix when the RSK algorithm is restricted to permutation, and hence the transposition of the matrix corresponds to inversion of the permutation. Burge gives a similar generalisation of the column insertion algorithm [3], which we refer to as Burge’s algorithm.

The symmetry property for the usual RS algorithm is normally discussed in the literature for the row insertion. However, in the permutation case, the output tableau pair for row insertion is simply the transposition of the pair for column insertion. Therefore, proofs of the symmetry property can be translated to column insertion instantly. In the matrix input case, the Burge and RSK algorithms are also closely related so that proofs can be extended from one to the other naturally.

For a proof of the symmetry property, one can see e.g. [9, 21, 23]. There are a few different approaches to deal with it. One is the Viennot diagram [25] which provides a nice geometric construction of the RS algorithms. In the matrix input case, there are two approaches which reduce to the Viennot diagram approach when restricted to permutations. One is the antichain or the inversion digraph construction of the RSK algorithm (which should extend naturally to the Burge algorithm) due to [13]; the other (for both RSK and Burge) is Fomin’s matrix-ball construction of [9].

Another method is the growth diagram technique due to Fomin [5, 6, 7, 8]. It can be generalised to the RSK and Burge algorithms. Greene’s theorem [11] gives a proof by showing the relation between the lengths of the longest subsequences of the input and the shape of the output. However, the growth diagram approach can be thought of as a fast construction to calculate these lengths.

Among all these techniques, the special structure of growth diagram can be extended to a class of algorithms what we will call branching algorithms, which include the \(q\)-weighted column and row insertion algorithms. Therefore, it is this approach which we use in this paper. For a simple description of the technique for usual row insertion see e.g. [23], whose column version will be shown in Sect. 3.

The rest of the paper is organised in the following way. In Sect. 2, we recall the insertion rule of a letter into a tableau for the usual RS algorithm (with column insertion). We describe it in a way that suits the growth diagram. In Sect. 3, we describe the insertion rule for a word and state the symmetry property for the RS algorithm applied to permutations. In Sect. 4 and 5, we describe the \(q\)-weighted insertion algorithm for letters and words in a way which is different, but equivalent to the definition given in [19]. In Sect. 6, we state and prove the symmetry property in the \(q\)-case. Finally, in Sect. 7, we prove the symmetry property for the row insertion algorithm and more generally branching algorithms.

## 2 Classical Robinson–Schensted algorithm

## 3 Symmetry property for the Robinson–Schensted algorithm

In this section, we define the insertion of words and recall Fomin’s formulation of the RS algorithm in terms of growth diagrams [5, 6, 7, 8] which we use to show the proof of the symmetry property, following the presentation given in [23, Sect. 7.13].

The RS algorithm was initially defined to take a permutation as an input and output a pair of *standard* tableaux with the same shape. In this case, we take the word identified by \(\sigma (1)\sigma (2)\sigma (3)\dots \sigma (n)\) as the input, which we also denote by \(\sigma \). A classical result of the algorithm is the symmetry property:

**Theorem 1**

*Proof*

- 1.
If there is an \(X\) in the box and \(\lambda =\mu ^1=\mu ^2\), then \(\nu =\lambda +\mathbf {e}_{l(\lambda )+1}\), that is \(\nu \) is obtained by adding a box that forms a new row itself at the bottom of \(\lambda \).

- 2.
If there is no \(X\) in the box and \(\lambda =\mu ^1\), then \(\nu =\mu ^2\).

- 3.
If there is no \(X\) in the box and \(\lambda =\mu ^2\), then \(\nu =\mu ^1\).

- 4.
If there is no \(X\) in the box and \(\mu ^1=\lambda +\mathbf {e}_i\), \(\mu ^2=\lambda +\mathbf {e}_j\) with \(i\ne j\), then \(\nu =\lambda +\mathbf {e}_i+\mathbf {e}_j=\mu ^1\cup \mu ^2\).

- 5.
If there is no \(X\) in the box and \(\mu ^1=\mu ^2=\lambda +\mathbf {e}_i\), then \(\nu =\lambda +\mathbf {e}_i+\mathbf {e}_{i'}\), where \(i'=\max (\{j\le i:\mu ^1_{j-1}>\mu ^1_j\}\cup \{1\}).\)

This way, all the vertices of the diagram can be labelled recursively given \(\emptyset \) as the boundary condition on the leftmost and bottom vertices. One could check that this is indeed equivalent to the definition in the previous section. Moreover, the transposition of the lattice diagram preserves the algorithm. That is: if we put \(X\)s in boxes \((\sigma (i),i)_{i\le n}\) rather than \((i,\sigma (i))_{i\le n}\), and label each vertex \((i,j)\) with what was labelled on \((j,i)\), we end up with the configuration that is the same as if we apply the rules 1 through 5. This immediately finishes the proof.

## 4 A \(q\)-weighted Robinson–Schensted algorithm

In a recent paper [19], a \(q\)-weighted RS algorithm was introduced. In this and the next section, we describe the algorithm in a different way from the definition in [19]. At the end of next section, it is obvious to see that:

**Proposition 2**

The algorithm described in this section for inserting a letter to a tableau and in the next section for inserting a word to an empty tableau is an equivalent reformulation of the \(q\)-weighted RS algorithm defined in [19].

When inserting a letter to a tableau, it outputs a weighted set of tableaux. To insert a \(k\) into \(P\), we start with \(\lambda ^k\), append a box to different possible rows no lower than \(k\), record the index of rows and for each one of these indices \(j_k\), we obtain a new \(k\)th shape \(\tilde{\lambda }^k=\lambda ^k+\mathbf {e}_{j_k}\) with weight \(w_0(k,j_k)\); then, we proceed to add a box to all possible rows in \(\lambda ^{k+1}\) no lower than \(j_k\) and obtain a weighted set of new \(k+1\)th shapes \(\{(\tilde{\lambda }^{k+1}=\lambda ^{k+1}+\mathbf {e}_{j_{k+1}},w_1(k+1,j_{k+1})):j_{k+1}\le j_k\}\); then, for each \(j_{k+1}\), we obtain the new \(k+2\)th shapes with weight \(w_1(k+2,j_{k+2})\) by adding a box to \(j_{k+2}\)th row in \(\lambda ^{k+2}\) and so on and so forth. We also prune all the 0-weighted branches.

This way we obtain a forest of trees whose roots are \(\tilde{\lambda }^k\)’s and leaves are \(\tilde{\lambda }^\ell \)’s with edges labelled by the weights. If we prepend \(\emptyset \prec \lambda ^1\prec \lambda ^2\prec \cdots \prec \lambda ^{k-1}\) to this forest, we obtain a tree with root \(\emptyset \), the first \(k\) levels each having one branch with one edge, which we label with weight 1. Then, for each leaf \(\tilde{\lambda }^\ell \), we obtain a unique tableau by reading its genealogy. We also associate this tableau with a weight which is the product of weights along the edges. By going through all leaves, we obtain a set of weighted tableaux, which is the output of \(q\)-inserting a \(k\) to \(P\).

All the nodes associated with the right column \(((i,0):0\le i\le \ell )\) form a tree, such that each \(\tilde{P}\) in the output corresponds to a genealogy of a node associated with \((1,\ell )\), whose weight \(I_k(P,\tilde{P})\) can be obtained as the product of the weights along the genealogical line of the tree.

## 5 Word input for the \(q\)-weighted Robinson–Schensted algorithm

*collection*we mean a vector of objects that allows repeats, as opposed to a

*set*. We merge triplets with the same tableau pair by adding up their weights. This way, we have obtained the weighted set of pairs of tableaux at time \(m+1\).

## 6 The symmetry property for the \(q\)-weighted RS algorithm with permutation input

When we take a permutation input \(\sigma \) which is identified as a word with distinct letters in the same way as in the usual RS algorithm, we also end up with a symmetry property, which is the main result of this paper:

**Theorem 3**

- 1.The box has an \(X\) in it, and \(\mu ^1\), \(\mu ^2\) are equal to \(\lambda \). Then, \(N\) consists of all possible partitions obtained by adding a box to \(\lambda \):The weights are$$\begin{aligned} N=\Lambda (\lambda ). \end{aligned}$$$$\begin{aligned} w(\mu ,\nu )={\left\{ \begin{array}{ll} q^{\lambda _j}-q^{\lambda _{j-1}},&{}\text {if }\nu =\lambda +\mathbf {e}_j\text { for some }j>1\\ q^{\lambda _1},&{}\text {if }\nu =\lambda +\mathbf {e}_1 \end{array}\right. },\quad \nu \in N. \end{aligned}$$
- 2.The box does not have an \(X\) in it and \(\mu ^1=\lambda \). Then, \(N\) is a singleton which is the same as \(\mu ^2\), and the weight is 1:$$\begin{aligned} w(\mu ,\nu )=1,\quad \nu \in N=\{\mu ^2\}. \end{aligned}$$
- 3.(The dual case of case 2) The box does not have an \(X\) in it and \(\mu ^2=\lambda \). Then, \(N\) is a singleton which is the same as \(\mu ^1\), and the weight is 1:$$\begin{aligned} w(\mu ,\nu )=1,\quad \nu \in N=\{\mu ^1\}. \end{aligned}$$
- 4.The box is empty. \(\mu ^1=\lambda +\mathbf {e}_i\) and \(\mu ^2=\lambda +\mathbf {e}_j\) for some \(i\ne j\). Then, \(N\) again only contains one element \(\mu ^1\cup \mu ^2\), with weight 1:$$\begin{aligned} w(\mu ,\nu )=1,\quad \nu \in N=\{\mu ^1\cup \mu ^2\}. \end{aligned}$$
- 5.The box is empty and \(\mu ^1=\mu ^2=\lambda +\mathbf {e}_i\,{:=}\,\mu \) for some \(i\). Then, \(N\) consists of all possible partitions that are obtained by adding a box to a row no lower than \(i\)th row of \(\mu \). That isThe weights are$$\begin{aligned} N=\Lambda ^i(\mu ). \end{aligned}$$$$\begin{aligned}&w(\mu ,\nu )\\&={\left\{ \begin{array}{ll} \frac{1-\,q^{\lambda _{i-1}-\lambda _i-1}}{1-\,q^{\lambda _{i-1}-\lambda _i}},&{}\nu =\mu +\mathbf {e}_i;\\ \frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}}q^{\lambda _j-\lambda _i-1}(1-\,q^{\lambda _{j-1}-\lambda _j}),&{}\nu =\mu +\mathbf {e}_j\text { for some }2\le j< i;\\ \frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}}q^{\lambda _1-\lambda _i-1},&{}\nu =\mu +\mathbf {e}_1. \end{array}\right. }\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad ,\nu \in N. \end{aligned}$$

*growth graph rule*as opposed to the

*insertion rule*described in Sect. 4 and 5. Below is a group of illustrations of all 5 cases, where \(\lambda \) is the southwest partition.

*Proof*

- 1.The box \((m,k)\) has an \(X\) in it. This means \(\sigma _m=k\). So, we are inserting a \(k\) to the tableau at time \(m\) and hence \(\lambda ^{k-1}=\tilde{\lambda }^{k-1}\), i.e. \(\lambda =\mu ^2\). Moreover, since \(\sigma \) is a permutation, we have \(\sigma _i\ne k\) for all \(i<m\), hence \(\lambda ^{k-1}=\lambda ^k\), i.e. \(\lambda =\mu ^1\) (condition of Case 1 satisfied). Any branch of \((\lambda ^{k-1},\lambda ^k,\tilde{\lambda }^{k-1})=(\lambda ,\lambda ,\lambda )\) is one of the \(\lambda +\mathbf {e}_j\)’s such that the weight \(w((\lambda ,\lambda ,\lambda ),\lambda +\mathbf {e}_j)=w_0(k,j)\ne 0\).So, the weights agree. All the pruned branches \(\lambda +\mathbf {e}_j\) with \(w_0(k,j)=0\) are exactly the \(\lambda +\mathbf {e}_j\)’s that are not partitions. Therefore, \(N\) is exactly the set of all branches of the triplet.$$\begin{aligned} w_0(k,j)&= f_0(j;\lambda ,\lambda )\prod _{p=j+1}^k(1-f_0(p;\lambda ,\lambda ))\\&= {\left\{ \begin{array}{ll} (1-q^{\lambda _{j-1}-\lambda _j})q^{\lambda _j}&{}\text { if }j>1\\ q^{\lambda _1}&{}\text { if }j=1. \end{array}\right. } \end{aligned}$$
- 2.
There is no \(X\) in \((i,k)\) for \(1\le i\le m\). This means \(\sigma _i\ne k\) for \(i\le m\). Since \(\sigma _i\ne k\) for \(i<m\), \(\lambda ^{k-1}=\lambda ^k\), i.e. \(\lambda =\mu ^1\) (condition of Case 2 satisfied). Moreover, since \(\sigma _m\ne k\), the triplet produces only one branch which is equal to \(\mu ^2\) with weight 1. This agrees with Case 2.

- 3.
There is no \(X\) in \((m,i)\) for \(1\le i\le k\). This means \(\sigma _m>k\). By the insertion rule, \(\lambda ^{k-1}=\tilde{\lambda }^{k-1}\), i.e. \(\lambda =\mu ^2\) (condition of Case 3 satisfied). Again, by the same rule, the triplet only produces one branch that equals \(\mu ^1\) with weight 1.

- 4.There is one \(X\) in each of \((t,k)\) and \((m,s)\) for some \(t<m\) and \(s<k\). Then, on the one hand, \(\sigma _t=k\) so \(\lambda ^k=\lambda ^{k-1}+\mathbf {e}_i\) for some \(i\), i.e. \(\mu ^1=\lambda +\mathbf {e}_i\). On the other hand, \(\sigma _m<k\) so \(\tilde{\lambda }^{k-1}=\lambda ^{k-1}+\mathbf {e}_j\) for some \(j\) and \(j_{k-1}=j\), i.e. \(\mu ^2=\lambda +\mathbf {e}_j\) (condition of Case 4 or Case 5 satisfied). Therefore, the branches of the triplet \((\lambda ,\lambda +\mathbf {e}_i,\lambda +\mathbf {e}_j)\) are in the form of \(\tilde{\lambda }^k=\lambda ^k+\mathbf {e}_{j_k}\) for \(j_k\le j\) with weightsIf \(j\ne i\) (condition of Case 4 satisfied), then \(f_1(j;\lambda ,\lambda +\mathbf {e}_i)=(1-q^{\lambda _{j-1}-\lambda _j})/(1-q^{\lambda _{j-1}-\lambda _j})=1\). Therefore, the branch has only one shape equal to \(\mu ^1+\mathbf {e}_j\) with weight 1. This agrees with Case 4. If \(j=i\) (condition of Case 5 satisfied), then by (2),$$\begin{aligned} \begin{aligned}&w((\lambda ,\lambda +\mathbf {e}_i,\lambda +\mathbf {e}_j),\lambda +\mathbf {e}_i+\mathbf {e}_{j_k})=w_1(k,j_k)\\&={\left\{ \begin{array}{ll} f_0(j_k;\lambda ,\lambda +\mathbf {e}_i)\left( \prod _{p=j_k+1}^{j-1}(1-f_0(p;\lambda ,\lambda +\mathbf {e}_i))\right) \\ \times (1-f_1(j;\lambda ,\lambda +\mathbf {e}_i))&{}\text { if }j_k<j\\ f_1(j;\lambda ,\lambda +\mathbf {e}_i)&{}\text { if }j_k=j \end{array}\right. } \end{aligned} \end{aligned}$$(2)So, the weights agree with Case 5. Moreover, all the pruned branches are exactly the \(\lambda +\mathbf {e}_i+\mathbf {e}_{j_k}\)’s that are not partitions. Therefore, \(N\) is indeed the set of all branches.$$\begin{aligned} w_1(k,j_k)= {\left\{ \begin{array}{ll} \frac{1-\,q^{\lambda _{i-1}-\lambda _i-1}}{1-\,q^{\lambda _{i-1}-\lambda _i}};&{}\text { if }j_k=i;\\ \frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}}q^{\lambda _j-\lambda _i-1}(1-\,q^{\lambda _{j-1}-\lambda _j});&{}\text { if }2\le j_k< i;\\ \frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}}q^{\lambda _1-\lambda _i-1};&{}\text { if }j_k=1. \end{array}\right. } \end{aligned}$$

The growth graphs defined above are a natural \(q\)-generalisation of the growth diagrams introduced by Fomin [5, 6, 7, 8] and reduce to Fomin’s growth diagrams when \(q\rightarrow 0\).

## 7 More insertion algorithms

*row insertion*. To row insert a \(k\) into a tableau \(P\), we again keep \(\lambda ^0\prec \lambda ^1\prec \cdots \prec \lambda ^{k-1}\) unchanged. Then, we append a box at the end of first row of \(\lambda ^k,\lambda ^{k+1},\ldots ,\lambda ^{k_1-1}\), where \(k_1\) is the smallest number in row 1 of \(P\) that is larger than \(k\), then we append a box at the end of second row of \(\lambda ^{k_1},\lambda ^{k_1+1},\dots ,\lambda ^{k_2-1}\), where \(k_2\) is the smallest number in row 2 of \(P\) that is larger than \(k_1\) and so on and so forth. More precisely, define:

The first is a well-known algebraic duality relation, which we now recall. For any tableau \(T\) and positive integers \(x,y\), denote by \(c_x\) (resp. \(r_y\)) the operation of column (resp. row) inserting \(x\) (resp. \(y\)), that is, \(c_xT\) (resp. \(r_yT\)) is the output tableau when column (resp. row) inserting \(x\) (resp. \(y\)) into \(T\). Denote by \((P_{\text {col}}(w),Q_{\text {col}}(w))\) (resp. \((P_{\text {row}}(w),Q_{\text {row}}(w))\)) the tableau pairs when applying column (resp. row) insertions to word \(w\), respectively. Thus, we have \(P_{\text {col}}(w)=c_{w_n}c_{w_{n-1}}\dots c_{w_1}\emptyset \) and \(P_{\text {row}}(w)=r_{w_n}r_{w_{n-1}}\dots r_{w_1}\emptyset \). Also, denote by \(w^r\) the inverse word of \(w\): \(w^r=(w_n,w_{n-1},\dots ,w_1)\). Moreover, denote by \(\text {ev}(Q)\) the evacuation operation on \(Q\), see e.g. [9, 21]. Then,

**Proposition 4**

This duality has a matrix input generalisation where \(Q_{\text {row}}(w^r)\) is obtained by a reverse sliding operation, see e.g. [9].

Another simple relation is between the original definitions. Row insertion was initially defined as an algorithm of inserting and bumping based on the ordering of integers. This definition turns into column insertion if one replaces all occurrence of ‘row’ by ‘column’ and replaces the strong order (greater than) with the weak order (greater than or equal to) due to the asymmetry of the ordering in rows and columns in the definition of a tableau. For these definitions, see e.g. [21].

There is also a third relation which is less well-known and can be seen if one describes the ‘dynamics’ of the column and row insertion algorithms in terms of the (Gelfand–Tsetlin) coordinates \((\lambda ^k_j)_{1\le j\le k\le \ell }\), where \(\lambda ^k\) is the \(k\)th shape of the \(P\)-tableau. In column insertion, we initialise \(j_{k-1}=j\). Then, in each step, we find the largest row index \(j_k\le j_{k-1}\) for a letter \(k\) such that \(\lambda ^{k-1}_{j_k-1}>\lambda ^k_{j_k}\), append a box to \(\lambda ^k_{j_k}\) and increase \(k\) by 1. In row insertion, we initialise \(k_0=k\), and in each step one we find the smallest letter \(k_j>k_{j-1}\) for a row index \(j\) such that \(\lambda ^{k_j}_j>\lambda ^{k_j-1}_j\), append a box to \(\lambda ^{k_{j-1}}_j,\dots ,\lambda ^{k_j-1}_j\) and increase \(j\) by 1. We shall refer to this relation as the *dynamical duality*.

In a recent paper [2], a \(q\)-weighted RS algorithm with row insertion was proposed. It is defined in a similar way as the \(q\)-column insertion and is related to the latter by an analogue of the dynamical duality described above.

In \(q\)-column insertion, we initialise \(j_{k-1}=k\). Then, in each step, we run over all row indices \(j_k\le j_{k-1}\) for a letter \(k\), with some weight append a box to \(\lambda ^k_{j_k}\) and increase \(k\) by 1. In \(q\)-row insertion, we initialise \(k_0=k\). In each step one, we run over all letters \(k_j>k_{j-1}\) for a row index \(j\) such that, with some weight, we append a box to \(\lambda ^{k_{j-1}}_j,\dots ,\lambda ^{k_j-1}_j\) and increase \(j\) by 1.

**Theorem 5**

*Proof*

- 1.
The box has an \(X\) in it, and \(\mu ^1\), \(\mu ^2\) are equal to \(\lambda \). Then, \(M\) consists of only one partition \(\lambda +\mathbf {e}_1\) with weight 1.

- 2.
The box does not have an \(X\) in it, and \(\mu ^1=\lambda \). Then, \(M\) consists of only one partition \(\mu ^2\) with weight 1.

- 3.
The box does not have an \(X\) in it, and \(\mu ^2=\lambda \). Then, \(M\) consists of only one partition \(\mu ^1\) with weight 1.

- 4.
The box is empty. \(\mu ^1=\lambda +\mathbf {e}_i\) and \(\mu ^2=\lambda +\mathbf {e}_j\) for some \(i\ne j\). Then, \(M\) again only contains one element \(\mu ^1\cup \mu ^2\) with weight 1.

- 5.The box is empty and \(\mu ^1=\mu ^2=\lambda +\mathbf {e}_i{:=}\mu \) for some \(i\). Then, \(M=\{\lambda +\mathbf {e}_i+\mathbf {e}_{i+1},\lambda +2\mathbf {e}_i\}\) with weight$$\begin{aligned} u((\lambda ,\lambda +\mathbf {e}_i,\lambda +\mathbf {e}_i),\lambda +\mathbf {e}_{i+1})&= {\left\{ \begin{array}{ll} 1-q,&{}\text { if }i=1;\\ \frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}},&{}\text { if }i>1. \end{array}\right. }\\ u((\lambda ,\lambda +\mathbf {e}_i,\lambda +\mathbf {e}_i),\lambda +2\mathbf {e}_i)&= {\left\{ \begin{array}{ll} q,&{}\text { if }i=1;\\ 1-\frac{1-\,q}{1-\,q^{\lambda _{i-1}-\lambda _i}},&{}\text { if }i>1. \end{array}\right. } \end{aligned}$$

Let us go back to the symmetry property. More generally, the symmetry property does not require very strong conditions on the insertion algorithms. Here, we give a sufficient condition for an insertion algorithm to have this property.

*branching insertion algorithm*as an algorithm that has the branching structure as in Fig. 1 when we insert a letter to a tableau. That is, there exists an

*initial branching weight function*\(w_0\), a

*high level weight function*\(w_1\) and a

*low level weight function*\(w_2\) such that when inserting a letter \(k\), the weight of a new \(i\)th shape is:

**Proposition 6**

- (i)
The insertion of \(k'\) into a tableau results in increment of one coordinate by 1 in \(\lambda ^{k'},\lambda ^{k'+1},\dots ,\lambda ^\ell \), while keep \(\lambda ^0,\lambda ^1,\dots ,\lambda ^{k'-1}\) unchanged, that is, the support of \(w_0((\lambda ^{i-1},\lambda ^i,\tilde{\lambda }^{i-1}),\tilde{\lambda }^i)\) and \(w_1((\lambda ^{i-1},\lambda ^i,\tilde{\lambda }^{i-1}),\tilde{\lambda }^i)\) are in \(\{\lambda ^i+\mathbf {e}_j:j\ge 1\}\) and \(w_2((\lambda ^{i-1},\lambda ^i,\tilde{\lambda }^{i-1}),\tilde{\lambda }^i)=\mathbb I_{\lambda ^i=\tilde{\lambda }^i}\),

- (ii)
\(w_1((\lambda ^{m-1},\lambda ^m,\lambda ^{m-1}+\mathbf {e}_i),\lambda ^m+\mathbf {e}_j)=\mathbb I_{i=j}\) if \(\lambda ^m_i=\lambda ^{m-1}_i\);

- (iii)
\(w_0((\lambda ,\lambda ,\lambda ),\lambda +\mathbf {e}_i)\) does not depend on the inserted letter;

- (iv)
\(w_1((\lambda ,\lambda +\mathbf {e}_i,\lambda +\mathbf {e}_i),\lambda +\mathbf {e}_i+\mathbf {e}_j)\) does not depend on the inserted letter,

*Proof*

(Sketch proof) We use the same notations \(\lambda ,\mu ^1,\mu ^2\) as in descriptions of growth graph rules such that they surround the box \((m,k)\) from the south and the west. The symmetry property is shown once we can construct a symmetric growth graph rule, where the relation of \(\lambda \), \(\mu ^1\) and \(\mu ^2\) and whether there is an \(X\) in the box, determines the location of the box in the permutation matrix, and vice versa.

Then, the 5 cases of the growth diagram rule are satisfied given these conditions: for Case 1, the equivalence between \((\lambda =\mu ^1=\mu ^2{\text { AND }}X)\) and \((\sigma _m=k)\) is given by (i) and (ii), and the branched shapes and weights are given by (iii); for Case 2, both the equivalence between \((\lambda =\mu ^1{\text { AND }}{\text { NOT }}X)\) and \((\sigma _s\ne k\forall s\le m)\) is given by (ii), and the singleton branching with weight \(1\) is also given by (ii); for Case 3, both the equivalence between \((\lambda =\mu ^2{\text { AND }}{\text { NOT }}X)\) and \((\sigma _m>k)\) is given by (i), and the singleton shape with weight \(1\) is also given by (i); for Case 4 and 5, the equivalence between \((\mu ^1=\lambda +\mathbf {e}_i{\text { AND }}\mu ^2=\lambda +\mathbf {e}_j{\text { AND }}{\text { NOT }}X)\) and \((\sigma ^{-1}_k<m{\text { AND }}\sigma _m<k)\) is given by (i) (ii), and the singleton shape with weight \(1\) in Case 4 is given by (ii) while the branched shapes and weights in Case 5 are given by (iv).

With this proposition, we can test different algorithms for the symmetry property. For example, a family of \(n!\) \(q\)-weighted algorithms is defined in [2], see Dynamics 3 in Sect. 6.5.3 and (8.5) in Sect. 8.2.1 in that paper. They are indexed by vectors \(h=(h_1,h_2,\dots ,h_n)\) with \(h_i\) being a positive integer less than or equal to \(i\). They are branching algorithms. The algorithm indexed by \(h=(1,1,\dots ,1)\) is the \(q\)-weighted row insertion, and as we have seen in Theorem 5 it has the symmetry property.

In [2] another family of ‘RSK-type’, \(q\)-weighted algorithms associated with vectors \(h\) called ‘\(q\)-Whittaker-multivariate “dynamics” with deterministic long-range interactions’ were introduced. We omit the definitions here and point interested readers to Sect. 8.2.2 of that paper. When \(h=(1,2,\dots ,n)\) and \(h=(1,1,\dots ,1)\), they also satisfy the conditions in Proposition 6 and thus enjoy the symmetry property.

## Notes

### Acknowledgments

The author would like to thank Neil O’Connell for guidance. Research of the author was supported by EPSRC Grant No. EP/H023364/1.

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