# The symmetric functions catalog

An overview of symmetric functions and related topics

2021-05-18

## Operations on Young tableaux

Here we cover some miscellaneous 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 $\evac^*$ is 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 $i^b(i+1)^a.$ It is easy to verify that $BK_i$ preserves the property of being a semistandard Young tableau.

Example (Bender–Knuth).

Here is the action of $BK_2,$ by showing the steps described above.

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.

### Jeu de taquin

There is a particular method to convert a skew SSYT to a non-skew 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.

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 on the order of the slides, and we denote the result by $\jdt(T).$

Example.

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

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.

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. In [Lemma 3.2, IV97], it is shown that jeu-de-taquin preserves the descent set. This can be generalized to semistandard Young tableaux.

In https://arxiv.org/pdf/2105.07819.pdf, the authors study a super version of JDT, which is related to a Littlewood–Richardson rule for the super Schur functions.

### 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.

Schützenberger showed that $\promotion^n(T)=T$ for any standard Young tableau $T$ of rectangular shape, with $n$ boxes. For SYT of shape $2 \times k,$ there is a bijection to non-crossing matchings, such that promotion is mapped 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.

We replace the $9$ with a bullet and perform slides:

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

See [PR17] for a proof that elements in $\SYT(m^n)$ with $m\geq n,$ which has order $n$ under promotion, is equinumerous with $\symS_n.$

Promotion can be defined on linear extension of posets, see R. Stanley, [Sta09]. A recent paper generalizes this to arbitrary poset labelings, see https://arxiv.org/pdf/2005.07187.pdf.

### k-Promotion

There is a generalization of promotion, acting on $\SSYT(\lambda,\cdot)$ with entries $\leq k.$ This was introduced by Schensted . 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 let the values in the skew shape $\lambda/\mu$ have value $k.$ 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 $\promotion_7(T).$

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

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

### Evacuation

Evacuation was introduced by M.-P. Schützenberger [Sch63], and is an involution $\evac : \SYT(\lambda) \to \SYT(\lambda).$ 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.

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

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.

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 [Remark 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.

For more on promotion and evacuation in other groups, see http://de.arxiv.org/pdf/1804.06736.pdf.

### RSK vs. dual RSK

Request! If there is a reference for this somewhere, please let me know.

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 of all, [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

$\pi \rskArrow (P,Q) \iff \rev(\pi) \rskArrow (P^t,\evac(Q^t)).$

From this property, we have that

$\begin{pmatrix} n & n-1 & \dotsc & 1 \\ v_n & v_{n-1} & \dotsc & v_1 \end{pmatrix} \rskArrow (P^t, n+1-(\evac_n(Q)^t)).$

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, says that

$\rev(w) \rskDualArrow P \iff \std(w) \rskArrow \std(P^t),$

so by standardization, we must have that the proposition holds for all biwords with distinct entries.

## Relation with crystal operators

### Crystals and with RSK

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 on row $1$ or $2$ on 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 shall 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 in according to 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{align} \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{align}

We also have the following relations with evacuation, see [Prop. 2.87, Shi05]. Here $T$ is a semi-standard 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

Warning! I have no good reference for the properties in this subsection, but it is fairly straightforward to prove these.

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.

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{align} \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{align}

### Relationship with the RSK III

Warning! I have no good reference for the properties in this subsection. but it is fairly straightforward to prove these.

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{align} \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{align}

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

## Other operators

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