Transitivity for Schutzenberger involutions on standard Young tableaux

Question

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?

Answer 1

Yes. It has been a while since I did this calculation but I think this is the gist of it:

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 $G_n$.

You can see this nicely using Speyer's cylindrical growth diagrams defined here.