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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