# Transitivity for Schutzenberger involutions on standard Young tableaux

Let $\lambda$ be a partition of $n$. Let $SYT(\lambda)$ denote the set of standard Young tableaux of shape $\lambda$.

For $i = 1, \dots, n$, let me define permutations $S_i$ of the set $SYT(\lambda)$. Let $S_n(T)$ be the Schutzenberger involution of $T$. For $i < n$, let $S_i(T)$ be the $i$th "partial" Schutzenberger involution. By this, I mean that we fix the part of $T$ containing $i+1, \dots, n$ and perform Schutzenberger involution of the part of $T$ containing $1, \dots, i$.

Berenstein and Kirillov studied these permutations, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf). They prove that these permutations give rise to the action of a certain group, called $G_n$, on $SYT(\lambda)$.

Question

Does the group $G_n$ act transitively on $SYT(\lambda)$? In other words, given two standard Young tableaux can I turn one into the other by applying a sequence of partial Schutzenberger involutions?

Let $s_{1q}$ be the permutation induced by the partial Schützenberger involution $S_q(\cdot)$. Then define the permutations $$s_{pq} := s_{1q} s_{1(q-p+1)} s_{1q}.$$ The $Q$-symbol of the RSK correspondence gives a bijection between words of length $n$ with shape $\lambda$ and $SYT(\lambda)$. One can check that on words the operators $s_{pq}$ when $p-q = 2$ induce the Knuth moves - they are in fact just the cactus operators, hence the choice of notation. Since the Knuth moves act transitively so does the group $G_n$.