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#Operations on Young tableaux

We collect several standard operations on tableaux. The reader is encouraged to read about the Robinson–Schensted–Knuth correspondence first.

Here is an overview of the notation:

  • \(\revCompl : \symS_n \to \symS_n\) via \(\pi \to n+1 - \rev(\pi),\) the reverse-complement operator.

  • \(\revCompl_m : [m]^n \to [m]^n\) is defined similarly as \(w \to m+1 - \rev(w).\)

  • \(BK_i\) is the Bender–Knuth involution.

  • \(\evac\) and \(\evac^*\) are evacuation and dual evacuation, respectively.

  • \(\promotion\) is promotion.

  • \(\jdt\) is the jeu-de-taquin sliding operator.

  • \(\ins\) is the operator that sends a word to its insertion tableau under row insertion.

We let \(\revCompl\) act on skew standard Young tableaux with \(n\) boxes by rotating the shape \(180^\circ\) and sending entry \(k\) to \(n+1-k.\)

#Bender–Knuth involutions

The Bender–Knuth involution \(BK_i\) is a map on \(\SSYT(\lambda/\mu, w)\) to \(\SSYT(\lambda/\mu, s_i w),\) where \(w\) is the weight of the tableau.

For \(T \in \SSYT(\lambda/\mu, w),\) ignore all entries not equal to \(i\) or \(i+1.\) Furthermore, ignore columns that contain both \(i\) and \(i+1.\) What remains is a disjoint set of rows; each row is a sequence of \(i\)s immediately followed by a sequence of \((i+1)\)s.

The map \(BK_i\) replaces each such sequence \(i^a(i+1)^b\) with \(i^b(i+1)^a.\) The resulting filling is again a semistandard Young tableau.

Example (Bender–Knuth).

Here is the action of \(BK_2,\) by showing the steps described above.

$1$ $1$ $1$ $1$ $1$ $1$ $2$ $2$ $2$ $2$ $3$ $ 2$ $2$ $2$ $2$ $2$ $2$ $3$         $ 3$ $4$ $4$ $4$ $4$                           $2$ $2$ $2$ $3$   $2$ $2$ $2$ $2$ $2$                                               $2$ $3$ $3$ $3$   $3$ $3$ $3$ $3$ $3$                                 $1$ $1$ $1$ $1$ $1$ $1$ $2$ $2$ $3$ $3$ $3$ $ 2$ $3$ $3$ $3$ $3$ $3$ $3$         $ 3$ $4$ $4$ $4$ $4$            

The weight of the first tableau is \((6,10,3,4)\) while the image has weight \((6,3,10,4).\)

The operations \(BK_i\) can be used to show that Schur polynomials are symmetric. Note, however, that the \(BK_i\) do not satisfy the braid relations! See the Lascoux–Schützenberger involutions instead for such a family of involutions.

�. Gutiérrez gives type \(B\) and type \(C\) analogues of the Bender–Knuth involutions [Gut24]. These give a combinatorial proof that the King and Sundaram tableau formulas for the symmetric polynomials of types \(B\) and \(C\) are invariant under the corresponding Weyl groups.

#Jeu de taquin

A standard method converts a skew SSYT to a straight shape via a series of slides, called jeu de taquin. The slides are defined as follows under the conditions that \(a \lt b\) and \(a \leq b,\) respectively. The boxes \(b\) and \(c\) may or may not be present.

$ \bullet $ $ a $       $ a $ $ \bullet$ $ b $ $ c $   $ \rightarrow $   $ b $ $ c$ $ \bullet $ $ b $       $ a $ $ b$ $ a $ $ c $   $ \rightarrow $   $ \bullet $ $ c$

Here, \(\bullet\) represents an empty square. The sliding procedure applied to a skew tableau \(T\) stops when a non-skew shape has been reached. It turns out that the result is independent of the order of the slides, and we denote the result by \(\jdt(T).\)

Example (Example of the sliding procedure).

The following is a sequence of slides applied to a skew tableau.

  $ \bullet $ $ 1 $ $ 1$   $ 2 $ $ 3$   $ 3 $ $ 3$       $ 1 $ $ \bullet $ $ 1$   $ 2 $ $ 3$   $ 3 $ $ 3$       $ 1 $ $ 1$ $ \bullet $ $ 2 $ $ 3$ $ 3 $ $ 3$     $ 1 $ $ 1$ $ 2 $ $ \bullet $ $ 3$ $ 3 $ $ 3$   $ \bullet $ $ 1 $ $ 1$ $ 2 $ $ 3 $ $ 3$ $ 3$     $1 $ $ \bullet $ $ 1$ $ 2 $ $ 3 $ $ 3$ $ 3$     $1 $ $ 1 $ $ \bullet$ $ 2 $ $ 3 $ $ 3$ $ 3$     $1 $ $ 1 $ $ 3$ $ 2 $ $ 3$   $ 3$    

Theorem (See [Cor. 2.3.19, But94]).

Let \(T\) be the skew tableau with \(w_i\) at position \((n+1-i,i).\) Then \(\jdt(T)\) is the same as the insertion tableau \(\ins(w).\)

In fact, row-insertion can be emulated via jeu-de-taquin.

Example (Row insertion as jeu-de-taquin).

For the word \(w=213,\) ordinary row insertion gives the following successive tableaux.

$2$ $1$ $2$ $1$ $3$ $2$  

The same tableau is obtained by placing the letters of \(w\) on the anti-diagonal skew shape from the theorem above and rectifying the first tableau below to the second.

    $3$   $1$   $2$     $1$ $3$ $2$  

For a skew SSYT \(T,\) \(\ins(\rw(T)) = \jdt(T).\) In other words, performing jeu-de-taquin gives the same result as row insertion of the reading word of \(T.\)

Recall the notion of descent for skew SYT. W. F. D. IV proves that jeu-de-taquin preserves the descent set [Lem. 3.2, IV97]. This can be generalized to semistandard Young tableaux.

N. Hage studies a super version of jeu de taquin, related to a Littlewood–Richardson rule for the super Schur functions [Hag22].

#Promotion

Warning! What is called promotion here is sometimes denoted \(\promotion^{-1}.\)

​

We define promotion, \(\promotion,\) on a (skew) standard Young tableau with \(n\) boxes by the composition \[\promotion(T) \coloneqq \left( BK_{n-1} \circ BK_{n-2} \circ \dotsm \circ BK_2 \circ BK_1 \right)(T).\] In other words, we swap 1 and 2 if they are not in the same row or column, then swap 2 and 3 if they are not in the same row or column, and so on.

M.-P. Schützenberger showed that \(\promotion^n(T)=T\) for any standard Young tableau \(T\) of rectangular shape, with \(n\) boxes. For SYT of shape \(2 \times k,\) a bijection to non-crossing matchings sends promotion to rotation; see this example of cyclic sieving.

Promotion has the following properties, where \(T\) is any skew standard Young tableau: \[\promotion(T)^t = \promotion(T^t) \quad \text{ and } \quad \promotion \circ \revCompl \circ T = \revCompl \circ \promotion^{-1} \circ T.\]

Alternatively, the inverse of promotion can be defined via jeu-de-taquin as follows. Given \(T \in \SYT(\lambda),\) replace the largest entry with a \(\bullet,\) and perform jeu-de-taquin slides until \(\bullet\) is in the \((1,1)\) corner. Add \(1\) to all entries in the tableau and finally replace the \(\bullet\) with a \(1.\) The result is \(\promotion^{-1}(T).\)

Example (Promotion via jeu-de-taquin).

Let \(T\) be the following tableau.

$1$ $3$ $4$ $5$ $2$ $6$ $8$   $7$ $9$    

We replace the \(9\) with a bullet and perform slides:

$1$ $3$ $4$ $5$ $2$ $6$ $8$   $7$ $\bullet$     $1$ $3$ $4$ $5$ $2$ $6$ $8$   $\bullet$ $7$     $1$ $3$ $4$ $5$ $\bullet$ $6$ $8$   $2$ $7$     $\bullet$ $3$ $4$ $5$ $1$ $6$ $8$   $2$ $7$    

The \(\promotion^{-1}(T)\) is therefore

$1$ $4$ $5$ $6$ $2$ $7$ $9$   $3$ $8$    

See [PR17] for a proof that elements in \(\SYT(m^n)\) with \(m\geq n,\) whose promotion orbit has order \(n,\) are equinumerous with \(\symS_n.\)

Promotion can be defined on linear extensions of posets, see R. Stanley [Sta09]. C. Defant and N. Kravitz generalize this to arbitrary poset labelings through promotion sorting [DK22]. C. Defant introduces toric promotion, a cyclic analogue of promotion for graph labelings [Def22]. For forests, the orbit structure has a particularly simple description.

#k-Promotion

There is a generalization of promotion, acting on \(\SSYT(\lambda,\cdot)\) with entries \(\leq k.\) This was introduced by M.-P. Schützenberger [Sch63, Sch72]. We follow [BMS14]. Note that we define the inverse here, as this is consistent with previous notation.

The \(k\)-promotion operator \(\promotion_k\) may be defined via jeu-de-taquin as follows. Given \(T \in \SSYT(\lambda,\cdot),\) remove all entries equal to \(1,\) such that a skew tableau is formed. Perform jeu-de-taquin slides until a straight shape \(\mu \subseteq \lambda\) is reached. Subtract \(1\) from all entries in the tableau and finally fill the cells of \(\lambda/\mu\) with \(k.\) The result is a semistandard tableau of shape \(\lambda.\)

Note that if \(T\) has content \((\alpha_1,\dotsc,\alpha_k),\) then \(\promotion_k(T)\) has content \((\alpha_2,\dotsc,\alpha_k,\alpha_1).\)

Example (k-promotion).

Let \(T\) be the following tableau. All entries are \(\leq 7,\) so we can compute \(\promotion_7(T).\)

$1$ $1$ $1$ $2$ $2$ $3$ $2$ $2$ $3$ $4$ $4$ $5$ $4$ $4$ $6$       $7$          

We replace the \(1\)s and obtain a skew shape, on which jeu-de-taquin is performed.

$\bullet$ $\bullet$ $\bullet$ $2$ $2$ $3$ $2$ $2$ $3$ $4$ $4$ $5$ $4$ $4$ $6$       $7$           $\bullet$ $\bullet$ $2$ $\bullet$ $2$ $3$ $2$ $2$ $3$ $4$ $4$ $5$ $4$ $4$ $6$       $7$           $\bullet$ $\bullet$ $2$ $2$ $3$ $\bullet$ $2$ $2$ $3$ $4$ $4$ $5$ $4$ $4$ $6$       $7$           $\bullet$ $\bullet$ $2$ $2$ $3$ $5$ $2$ $2$ $3$ $4$ $4$ $\bullet$ $4$ $4$ $6$       $7$           $\bullet$ $2$ $2$ $2$ $3$ $5$ $2$ $\bullet$ $3$ $4$ $4$ $\bullet$ $4$ $4$ $6$       $7$           $\bullet$ $2$ $2$ $2$ $3$ $5$ $2$ $3$ $4$ $4$ $\bullet$ $\bullet$ $4$ $4$ $6$       $7$           $2$ $2$ $2$ $2$ $3$ $5$ $\bullet$ $3$ $4$ $4$ $\bullet$ $\bullet$ $4$ $4$ $6$       $7$           $2$ $2$ $2$ $2$ $3$ $5$ $3$ $4$ $4$ $4$ $\bullet$ $\bullet$ $4$ $6$ $\bullet$       $7$          

We then subtract \(1\) from all entries, and replace the \(\bullet\) with \(k=7\) in order to find \(\promotion_k(T),\)

$1 $ $ 1$ $1$ $1$ $2$ $4$ $ 2$ $3$ $3$ $3 $ $ 7 $ $ 7$ $ 3 $ $ 5 $ $ 7$       $ 6$          

#Evacuation

The evacuation operator was introduced by M.-P. Schützenberger [Sch63], and is an involution \(\evac : \SYT(\lambda) \to \SYT(\lambda).\) It is a special case of the Lusztig involution on crystals, see e.g. [MPS18].

It can be defined in several ways, the following is from [But94].

Place \(T \in \SYT(\lambda)\) in a tight rectangle, and replace entry \(j\) with \(n+1-j.\) Rotate the rectangle \(180^\circ\) and perform jeu-de-taquin slides on the resulting (skew) shape until a standard Young tableau is obtained. In other words, \(\evac(T) = \jdt \circ \revCompl(T).\)

Example (Evacuation).

Evacuation \(\evac\) is computed step by step as described above.

$1$ $3$ $4$ $8$ $2$ $5$ $6$   $7$ $9$     $9$ $7$ $6$ $2$ $8$ $5$ $4$   $3$ $1$         $1$ $3$   $4$ $5$ $8$ $2$ $6$ $7$ $9$ $1$ $3$ $5$ $8$ $2$ $4$ $9$   $6$ $7$    

As a second example, taken from [PW11] and adapted to our notation, \(\evac\) sends the first SYT below to the second.

$1$ $3$ $8$ $2$ $4$   $5$ $9$   $6$ $10$   $7$     $1$ $4$ $9$ $2$ $5$   $3$ $6$   $7$ $10$   $8$    

Alternatively, evacuation on \(T \in \SYT(\lambda)\) can be computed using row-insertion as follows. \[\evac(T) = \ins \circ \revCompl \circ \rw(T)\]

We may also define evacuation via Bender–Knuth operators, now acting on \(\SSYT(\lambda,d):\) \[\evac_d \coloneqq (BK_1) \circ (BK_2 \circ BK_1) \circ (BK_3 \circ BK_2 \circ BK_1) \circ \dotsb \circ (BK_{d-1} \circ BK_{d-2} \circ \dotsb \circ BK_2 \circ BK_1)\] Furthermore, \(\evac_d : \SSYT(\lambda,d) \to \SSYT(\lambda,d)\) is also defined via \[\evac_d(T) = \jdt \circ \revCompl_d(T) = \ins \circ \revCompl_d \circ \rw(T).\]

There is also the notion of dual evacuation, \(\evac^*,\) which can be defined via promotion and ordinary evacuation as \(\evac^*(T) \coloneqq \promotion^n \circ \evac(T),\) see [Sta09] for details. M. Chmutov, G. Frieden, D. Kim, Joel B. Lewis, and E. Yudovina construct an affine evacuation operator on tabloids [CFKL+22]. The fixed points of the affine evacuation involution are counted by a Green polynomial evaluated at \(q=-1.\)

As a map \(\evac_d^* : \SSYT(\lambda,d) \to \SSYT(\lambda,d),\) it can be defined via Bender–Knuth operators: \[\evac_d^* \coloneqq (BK_{d-1}) \circ (BK_{d-2} \circ BK_{d-1}) \circ (BK_{d-3} \circ BK_{d-2} \circ BK_{d-1}) \circ \dotsb \circ (BK_{1} \circ BK_2 \circ \dotsb \circ BK_{d-2} \circ BK_{d-1}).\]

We have the following properties of evacuation and promotion on SYT with \(n\) boxes. \[\evac^2 = (\evac^*)^2 = id \qquad \promotion \evac = \evac \promotion^{-1} \qquad \promotion^n = \evac \evac^*\]

Problem (See [Rem. 2.12, PW11]).

Can \(\evac^*\) be described using RSK, as with \(\evac\)? That is, is there some nice involution \(\revCompl^*(w)\) on words such that \(\revCompl^*(w) \rskArrow (\evac^* \circ P, \evac^* \circ Q)\)?

Proposition (RSK and evacuation).

Let \(w \in [d]^n,\) then \[w \rskArrow (P,Q) \iff \revCompl(w) \rskArrow (\evac_n \circ P, \evac \circ Q).\] For dual RSK, we have \[w \rskDualArrow (P,Q) \iff \revCompl(w) \rskDualArrow (\evac_n \circ P, \evac \circ Q).\] Of course, here one needs to transpose \(P\) before and after applying \(\evac_n\) as \(P^t\) is a semistandard Young tableau.

S. Pfannerer, M. Rubey, and B. Westbury study promotion on oscillating and alternating tableaux, relating it to rotation of matchings and permutations [PRW20].

#RSK vs. dual RSK

Proposition (RSK, dual RSK and evacuation).

Let \(\left(\begin{smallmatrix} u \\ v \end{smallmatrix} \right)\) be a biword with no repeated entries, lexicographically sorted. Suppose \[\begin{pmatrix} u \\ v \end{pmatrix} \rskArrow (P,Q).\] Then, by choosing \(m\) sufficiently large, \[\begin{pmatrix} \revCompl_m \circ u \\ \rev(v) \end{pmatrix} \rskDualArrow (P^t,\evac_m(Q^t)).\]

Proof

First, [Prop. 2.3.14, But94] states that for any word, \[w \rskArrow P \iff \rev(w) \rskDualArrow P^t.\] We can therefore be sure that the insertion tableaux are correct.

Suppose now that \(v\) is a permutation and that \[\begin{pmatrix} 1 & 2 & \dotsc & n \\ v_1 & v_2 & \dotsc & v_n \end{pmatrix} \rskArrow (P,Q).\] A property of RSK says that \[\pi \rskArrow (P,Q) \iff \rev(\pi) \rskArrow (P^t,\evac(Q^t)).\] Thus \[\begin{pmatrix} n & n-1 & \dotsc & 1 \\ v_n & v_{n-1} & \dotsc & v_1 \end{pmatrix} \rskArrow (P^t, n+1-(\evac_n(Q)^t)).\] Note that entries in \(Q\) must be complemented, as the top row is decreasing.

Furthermore, the insertion algorithms imply that for permutations \(\pi,\) \(\pi \rskArrow (P,Q)\) if and only if \(\pi \rskDualArrow (P,Q).\) Hence, we can conclude that \[\begin{pmatrix} n & n-1 & \dotsc & 1 \\ v_n & v_{n-1} & \dotsc & v_1 \end{pmatrix} \rskDualArrow (P^t, n+1-(\evac_n(Q)^t)).\] For \(v\) still being a permutation, \[\begin{pmatrix} u \\ v \end{pmatrix} \rskArrow (P,Q) \iff \begin{pmatrix} \revCompl_m \circ u \\ \rev(v) \end{pmatrix} \rskDualArrow (P^t, \evac_m(Q)^t).\] Hence, the recording tableaux agree whenever \(v\) has distinct entries.

Further properties of RSK say that \[\rev(w) \rskDualArrow P \iff \std(w) \rskArrow \std(P^t),\] so by standardization, the proposition holds for all biwords with distinct entries.

#Relation with crystal operators

The operators above interact well with the type \(A\) crystal operators on words and tableaux. The basic principle is that RSK transports crystal operators on one row of a biword to crystal operators on one of the two tableaux, while the other tableau is unchanged. Evacuation reverses the crystal arrows, exchanging \(\cryse_i\) with \(\crysf_{n-i}.\)

#Crystals and RSK

D. Rush uses promotion, evacuation, and cyclic sieving in his study of plethysm coefficients [Rus18].

We have the following properties of crystals and RSK. These are phrased for the operator \(\cryse_i,\) but any of \(\cryse_i,\) \(\crysf_i\) or \(\cryss_i\) can be used.

Caveat! In order to apply an operator to row \(1\) or \(2\) of a biword, one should first sort the columns according to the entries in the other row, followed by application of the operator.

Example (Applying \(\cryse_1\) on a biword).

We now illustrate how to apply \(\cryse_1\) on the top row of a biword. Let us start with the biword \(W:\) \[\begin{pmatrix} 1&1&2&2&2&3 \\ 1&3&1&1&2&2 \end{pmatrix}\] We first sort the columns according to the second row: \[\begin{pmatrix} 1&2&2&2&3&1 \\ 1&1&1&2&2&3 \end{pmatrix}\] We then apply \(\cryse_1\) on the top row: \[\begin{pmatrix} 1&1&2&2&3&1 \\ 1&1&1&2&2&3 \end{pmatrix}\] The biword is then re-sorted according to the top row: \[\begin{pmatrix} 1&1&1&2&2&3 \\ 1&1&3&1&1&2 \end{pmatrix}\]

#Relationship with the RSK I

For proofs of the following statements, see [Thm. 5.5.1, Lot02].

Let \(w \in [d]^n.\) Then \[w \rskArrow (P,Q) \qquad \iff \qquad \cryse_i \circ w \rskArrow (\cryse_i \circ P,Q).\] This follows from the following more general statement. Let \(W = (w_1,w_2)^T\) be a biword. Then the following are equivalent. \[\begin{aligned} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} & \rskArrow (P,Q) \\ \begin{pmatrix} \cryse_i \circ w_1 \\ w_2 \end{pmatrix} & \rskArrow (P, \cryse_i \circ Q) \\ \begin{pmatrix} w_1 \\ \cryse_i \circ w_2 \end{pmatrix} & \rskArrow ( \cryse_i \circ P, Q) \end{aligned}\]

We also have the following relations with evacuation, see [Prop. 2.87, Shi05]. Here \(T\) is a semistandard tableau with \(n\) boxes.

  • \(\crysf_i(\evac_n(T)) = \evac_n( \cryse_{n-i}(T) )\)

  • \(\cryse_i(\evac_n(T)) = \evac_n( \crysf_{n-i}(T) )\)

  • \(\cryss_i(\evac_n(T)) = \evac_n( \cryss_{n-i}(T) )\)

Note that if \(\equiv\) denotes Knuth equivalence, then [Thm. 5.5.1, Lot02] \[w \equiv w' \iff \cryse_i(w) \equiv \cryse_i(w') \iff \crysf_i(w) \equiv \crysf_i(w') \iff \cryss_i(w) \equiv \cryss_i(w').\]

#Relationship with the dual RSK

Marc A. A. van Leeuwen’s double-crystal viewpoint gives a reference for the compatibility statements in this subsection [Lee06]. In that paper, horizontal and vertical crystal operations on binary matrices commute, and the resulting matrix decomposition is related to the dual RSK correspondence; see especially [Prop. 3.3.4, Lee06]. We use the dual RSK convention from the RSK page; van Leeuwen notes that the common variants differ by transposition and Schützenberger duality.

​

Let \(W = (w_1,w_2)^T\) be a biword obtained from a binary matrix. To apply \(\cryse_i\) on the top row of \(W,\) sort the biword primarily in an increasing fashion according to the bottom row, and secondarily in a decreasing fashion in the top row.

This is the row-word form of applying the corresponding matrix crystal operator in the horizontal direction.

To apply \(\cryse_i\) on the bottom row of \(W,\) sort the biword primarily in a decreasing fashion according to the top row, and secondarily in a decreasing fashion in the bottom row. Then the following are equivalent. \[\begin{aligned} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} & \rskDualArrow (P,Q) \\ \begin{pmatrix} \cryse_i \circ w_1 \\ w_2 \end{pmatrix} & \rskDualArrow (P, \cryse_i \circ Q) \\ \begin{pmatrix} w_1 \\ \cryse_i \circ w_2 \end{pmatrix} & \rskDualArrow (\cryse_i \circ P, Q) \end{aligned}\]

#Relationship with the RSK III

The Burge case is covered by the same double-crystal framework. In [Prop. 3.3.2, Lee06], the integral matrix decomposition is identified with the Burge correspondence, so the statement below is the Burge specialization of the horizontal/vertical crystal compatibility.

​

We now examine the third class of RSK where we use Burge biwords, where the biword entries are primarily sorted according to the top entry, but secondarily in a decreasing fashion on the bottom entry. We still perform the usual RSK bumping algorithm on Burge biwords.

To perform \(\cryse_i\) on the bottom row of a Burge word we simply apply the operator. To apply \(\cryse_i\) on the top row, sort the biword primarily in a decreasing fashion according to the bottom row, and secondarily in a decreasing fashion in the top row as well. Then apply \(\cryse_i\) to the top row and rearrange columns into Burge order again.

Let \(W = (w_1,w_2)^T\) be a Burge biword corresponding to a binary matrix. Then the following are equivalent. \[\begin{aligned} \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} & \rskbArrow (P,Q) \\ \begin{pmatrix} w_1 \\ \cryse_i \circ w_2 \end{pmatrix} & \rskbArrow (\cryse_i \circ P, Q) \\ \begin{pmatrix} \cryse_i \circ w_1 \\ w_2 \end{pmatrix} & \rskbArrow (P, \cryse_i \circ Q) \\ \end{aligned}\]

This version of RSK is studied extensively by O. Azenhas, see [Aze06], in particular in relationship with key polynomials.

#Other operators

There are also the cyclage and catabolism operations on semistandard Young tableaux.

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