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First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling of the boxes of (the Young diagram of) $\lambda$ with positive integers such that entries strictly increase down columns and weakly increase along rows. For such a tableau $T$ we define $\mathbf{x}^{T} := \prod_{i} x_i^{a_i(T)}$ where $a_i(T):=\#\textrm{$i$'s in $T$}$. For $\lambda$ a partition with at most $m$ parts, the generating function $$ s_{\lambda}(x_1,\ldots,x_m) := \sum_{T} \mathbf{x}^{T},$$ where the sum is over all semistandard Young tableaux of shape $\lambda$ with entries in $\{1,\ldots,m\}$, is the character of the irreducible, finite-dimensional representation $V^{\lambda}$ of $\mathfrak{gl}_m$ with highest weight $\lambda$. This is all classical.

Now, the characters of $\mathfrak{gl}_m$ are invariant under the action of the Weyl group of $\mathfrak{gl}_m$, a.k.a. the symmetric group $\mathfrak{S}_m$. Bender and Knuth defined certain operators on the set of semistandard tableaux, now called Bender-Knuth involutions, which allow one to see this symmetry combinatorially (the involutions swap the quantities $a_i(T)$ and $a_{i+1}(T)$).

King (see paper cited below) defined tableaux for the symplectic Lie algebra. Namely, for a partition $\lambda$ with at most $n$ rows, a symplectic tableau of shape $\lambda$ is a filling of the boxes of $\lambda$ with the symbols $\overline{1}<1<\overline{2}<2<\cdots <\overline{n}<n$ (with the symbols totally ordered that way) such that:

  • the entries strictly increase down columns and weakly increase down rows (semistandard condition);
  • entries $i$ and $\overline{i}$ do not appear below row $i$ (symplectic condition).

For such a tableau $T$ we define $\mathbf{x}^{T} := \prod_{i} x_i^{a_i(T)}$ where $a_i(T):=\#\textrm{$i$'s in $T$} - \#\textrm{$\overline{i}$'s in $T$}$. Then King showed the generating function $$ sp_{\lambda}(x_1,\ldots,x_m) := \sum_{T} \mathbf{x}^{T},$$ where the sum is over all symplectic tableaux of shape $\lambda$, is the character of the irreducible, finite-dimensional representation $V^{\lambda}$ of $\mathfrak{sp}_{2n}$ with highest weight $\lambda$.

Now, $sp_{\lambda}(x_1,\ldots,x_m)$ must be invariant under the action of the Weyl group of $\mathfrak{sp}_{2n}$, i.e., the hyperoctahedral group $\mathfrak{S}_2 \wr\mathfrak{S}_n$. In other words, $sp_{\lambda}(x_1,\ldots,x_m)$ is invariant under permuting and negating the exponents of the $x_i$.

Question: Are there Bender-Knuth-like involutions for symplectic tableaux that allow one to see this symmetry combinatorially?

I thought this should be well-known, but googling "symplectic Bender-Knuth" did not seem to turn up anything useful. Note that for negating $a_i(T)$, I believe the usual Bender-Knuth involution should work; but for swapping the values of $a_{i}(T)$ and $a_{i+1}(T)$, the symplectic condition causes problems if one tries to naively apply the usual Bender-Knuth involution.

King, R. C., Weight multiplicities for the classical groups, Group theor. Meth. Phys., 4th int. Colloq., Nijmegen 1975, Lect. Notes Phys. 50, 490-499 (1976). ZBL0369.22018.

EDIT:

In case it's helpful, let me mention another way to think about Bender-Knuth involutions, using Gelfand-Tsetlin patterns. Recall that a Gelfand-Tsetlin pattern of size $n$ is a triangular array $$\begin{array}{c c c c c} a_{1,1} & a_{1,2} & a_{1,3} & \cdots & a_{1,n}\\ & a_{2,2} & a_{2,3} & \cdots & a_{2,n} \\ & & \ddots & \cdots & \vdots \\ & & & a_{n-1,n} & a_{n,n} \\ & & & & a_{n,n} \end{array}$$ of nonnegative integers that is weakly decreasing in rows and columns. There is a well known bijection between semistandard Young tableaux of shape $\lambda = (\lambda_1,\ldots,\lambda_n)$ with entries $\leq n$ and GT patterns with $0$th (i.e., main) diagonal $(a_{1,1},a_{2,2},\ldots,a_{n,n})=(\lambda_1,\ldots,\lambda_n)$. Moreover, as shown in Proposition 2.2 of the paper of Berenstein and Kirillov below, the $i$th Bender-Knuth involution for $i=1,\ldots,n-1$ acting on the set of these tableaux can be realized by toggling (in a piecewise-linear manner) along the $i$th diagonal of the corresponding GT pattern.

For symplectic tableaux, there is also a GT pattern-like model. Namely, the $n$-symplectic tableaux of shape $\lambda=(\lambda_1,\ldots,\lambda_n)$ are in bijection with ``trapezoidal'' arrays $$\begin{array}{c c c c c c c c} a_{1,1} & a_{1,2} & a_{1,3} & \cdots & \cdots & a_{1,2n-2} & a_{1,2n-1} & a_{1,2n} \\ & a_{2,2} & a_{2,3} & \cdots & \cdots & a_{2,2n-2} & a_{2,2n-1} \\ & & a_{3,3} & \cdots & \cdots & a_{3,2n-2} \\ & & & \vdots & \vdots \\ & & & a_{n,n} & a_{n,n+1} \end{array}$$ of nonnegative integers that are weakly decreasing in rows and columns, and where again we have $(a_{1,1},a_{2,2},\ldots,a_{n,n})=(\lambda_1,\ldots,\lambda_n)$; see for instance Lemma 2 of the paper of Proctor cited below. It might be reasonable to try to realize the symplectic Bender-Knuth operations by toggling along diagonals of these trapezoidal arrays; but note that this trapezoid shape has $2n$ diagonals, which is a lot more than the $n$ involutions we expect to generate the relevant hyperoctahedral group.

Kirillov, A. N.; Berenstein, A. D., Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, St. Petersbg. Math. J. 7, No. 1, 77-127 (1996); translation from Algebra Anal. 7, No. 1, 92-152 (1995). ZBL0848.20007.

Proctor, Robert A., Shifted plane partitions of trapezoidal shape, Proc. Am. Math. Soc. 89, 553-559 (1983). ZBL0525.05007.

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    $\begingroup$ A symplectic analogue of the Bender-Knuth involution appears as Theorem 6.12 in the Ph.D. thesis of Sheila Sundaram, available at library.mit.edu/F/…. $\endgroup$ Jun 14, 2020 at 13:58
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    $\begingroup$ @RichardStanley: I think the argument in Theorem 6.12 is incomplete. (Note also she uses the convention $1<\overline{1}<2<\overline{2}<\cdots<n<\overline{n}$, but this is ok since we can swap $i$ and $\overline{i}$ in the BK way as I mentioned.) Imagine a one-column tableau with first entry $\overline{1}$ and second entry $2$. Sundaram does not really specify what to do here to swap $\alpha_1$ and $\alpha_2$. $\endgroup$ Jun 14, 2020 at 14:08
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    $\begingroup$ @SamHopkins Are you looking to prove CSP on King tableaux? Sounds like a fun project! We should collaborate on some paper regarding CSP - I have a few conjectures waiting to be solved :) $\endgroup$ Jun 14, 2020 at 19:20
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    $\begingroup$ @PerAlexandersson: I did not ask this question with any particular application in mind. It's just that after SSYT for the general linear group, symplectic tableaux are the next 'nicest' tableaux modeling the characters of classical groups, and I thought this should be something known. $\endgroup$ Jun 14, 2020 at 19:33
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    $\begingroup$ @SamHopkins: I don't understand the problem with Sheila Sundaram's proof. The hyperoctahedral group is generated by $(1,\bar 1)$ and the transpositions $(i,i+1)$. To swap $i$ and $i+1$, we can proceed as with $GL_n$ semistandard tableaux. To swap $1$ and $\bar 1$, the same trick works. $\endgroup$ Jun 14, 2020 at 20:48

2 Answers 2

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Let the hyperoctahedral group $\mathbb{S}_n\wr\mathbb{S}_2$ act naturally on the set $\{1, 2, ..., n\} \cup \{1', 2', ..., n'\}$. Then, we can say $\mathbb{S}_n\wr\mathbb{S}_2$ is generated by the transposition $(1\quad 1')$ and the permutations $(i\quad i\!+\!1)(i'\quad i\!+\!1')$ for $i = 1, ..., n-1$. We want to show that the number of King's tableaux (or King's patterns) of a fixed shape $\lambda$ and a given weight $x^\alpha$ is equal to that of weight $s.x^\alpha = x^{s(\alpha)}$ for all generators $s$. In particular, it is enough to define an action of $\mathbb{S}_n\wr\mathbb{S}_2$ on the set of King's tableaux such that $s.(\text{weight}(T)) = \text{weight}(s.T)$ and such that $s^2.T = T$ for all tableaux $T$ and generators $s$.

For $s = (1\quad 1')$, one can use type A Bender--Knuth involutions. To do this, one relabels the tableau from the alphabet $1<1'<2<2'<\cdots<n<n'$ to the alphabet $1<2<\cdots<2n$, applies the 1st Bender--Knuth involution, and relabels back.

For $s = (i\quad i\!+\!1)(i'\quad i\!+\!1')$, we first decompose the generator as a product of simple transpositions, $s = (i'\quad i\!+\!1)(i\!+\!1\quad i\!+\!1')(i\quad i')(i'\quad i\!+\!1)$. One now performs the four associated type A Bender--Knuth involutions. The result might not be a King tableau. It is shown in my MSc thesis that postcomposition with a suitable "rectifying" map gives a King tableau, and that this action of $s$ is involutory.

The text is available in my webpage (Proposition 5.9 and Appendix A).

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    $\begingroup$ This is really nice! Thank you for taking the time to work through the details of this in your thesis. I guess the combinatorics are slightly more involved than what you might expect at first blush. $\endgroup$ Jul 25, 2023 at 14:20
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For any semisimple Lie algebra $ \mathfrak g $ and any crystal $ B $ of a $\mathfrak g$-representation, we have an action of the cactus group $ C_{\mathfrak g} $ on $ B $. We have a surjective group homomorphism onto the Weyl group $ C_{\mathfrak g} \rightarrow W_{\mathfrak g} $. The action of $ C_{\mathfrak g} $ on $ B$ makes manifest the Weyl group symmetry in the weight multiplicities of the representation attached to $ B $. This is explained in our paper https://arxiv.org/abs/1708.05105.

In Halacheva's paper, https://arxiv.org/abs/2001.02262, she proved that for $ \mathfrak g = \mathfrak{sl}_n $, the Bender-Knuth moves generate this action of the cactus group (see also Chmutov-Glick-Pylyavskyy https://arxiv.org/abs/1609.02046). However, the Bender-Knuth moves are not the actions of the usual generators of the cactus group.

So for $ \mathfrak{sp}_{2n} $, we have the cactus group action (which is generated by involutions) on tableaux which makes manifest the Weyl group symmetry. The only remaining question is whether there are certain elements of the cactus group whose action looks like "Bender-Knuth" on these symplectic tableaux.

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    $\begingroup$ Yes, it's the last step of what does the action look like on tableaux that I'm interested in. $\endgroup$ Jun 14, 2020 at 13:33
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    $\begingroup$ My first reaction was that I'd expect a Bender-Knuth / Cactus group action on the De Concini-Procesi / Kashiwara-Nakashima symplectic tableaux, similar to arxiv.org/abs/1609.02046. $\endgroup$ Jun 14, 2020 at 13:40
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    $\begingroup$ I don't know if it's useful, but a crystal structure on King tableaux has recently been constructed in: arxiv.org/abs/1910.04459 $\endgroup$ Jun 14, 2020 at 13:57
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    $\begingroup$ Does this paper answer your question: arxiv.org/abs/2104.11799 $\endgroup$ May 20, 2021 at 14:07
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    $\begingroup$ @JoelKamnitzer: I only just saw this comment now. Interesting paper, thanks for making me aware of it! However, I don't think it quite answers my question, because the model of symplectic tableaux is not the King tableaux I am interested in (look at Definition 2.1). $\endgroup$ Dec 18, 2021 at 4:01

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