Proof that the Lascoux-Schützenberger involutions satisfies the braid-relations

Question

I am interested in the Lascoux-Schützenbereger involutions $\theta_i$, defined on semistandard Young tableaux. See this paper for definitions, page 4. These involutions satisfy the braid relations:

(i) $ \theta_i^2 = \text{Id}$

(ii) $\theta_i\theta_j\theta_i=\theta_j\theta_i\theta_j$, for $\mid i-j \mid=1$

(iii) $\theta_i\theta_j=\theta_j\theta_i$, for $\mid i-j\mid > 1$.

It is clear that $(i)$ and $(iii)$ holds. However, I do not know how to prove $(ii)$ and I cannot seem to find a reference for this fact that is not the original article by Lascoux and Schützenberger, which is in French (my French is not that great, unfortunately). Does anyone know another reference and/or a proof of this?

EDIT: Changed notation from $s_i$ to $\theta_i$ since this is this notation used in the link.

Answer 1

I have found another reference: Theorem $5.6.3$ in Algebraic Combinatorics on Words by M. Lothaire.

Answer 2

This is one of those cases where the margin is too small for everyone. Another proof is found in Marc A. A. van Leeuwen, Double crystals of binary and integral matrices, Electron. J. Combin., 13(1):Research Paper 86, 93, 2006. He uses different notations and works in a somewhat different setting; for details about how to derive your claim from van Leeuwen's work, see Remark 6.6 in Erik Aas, Darij Grinberg, Travis Scrimshaw, Multiline queues with spectral parameters, arXiv:1810.08157v1.

(I also have a different proof, and no time to write it up... It's on my tenure-track to-do list.)