Bender-Knuth involutions for symplectic (King) tableaux

Question

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.

Answer 1

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).

Answer 2

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.