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](https://en.wikipedia.org/wiki/Bender%E2%80%93Knuth_involution), 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](https://zbmath.org/?q=an:0369.22018).


**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](https://zbmath.org/?q=an:0848.20007).


*Proctor, Robert A.*, [**Shifted plane partitions of trapezoidal shape**](http://dx.doi.org/10.2307/2045516), Proc. Am. Math. Soc. 89, 553-559 (1983). [ZBL0525.05007](https://zbmath.org/?q=an:0525.05007).



Bender-Knuth involutions for symplectic (King) tableaux
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](https://sites.google.com/view/gutierrez-caceres/) (Proposition 5.9 and Appendix A).


