Let $S_n$ be the symmetric group and let $\lambda$ be a partition of $n$ and let $S$ be the set of all standard young tableaux of shape $\lambda$. Is there a well defined action of $S_n$ on the set $S$ ? Permuting entries of a standard tableau may not give a standard tableau. May be some kind of rearrangement needed after the permutation action.
Such an action was defined by Lascoux and Schützenberger in their paper Le monoïde plaxique, http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1981-1PlaxiqueNaples.pdf. Another reference is Theorem 1.5 of https://arxiv.org/pdf/q-alg/9709010.pdf.
To give an alternative reference, see the intro in Crystals For Dummies by M. Shimozono, where you want to be looking at the action $s_i$.
This is strongly related to the action as R. Stanley refer to, and also extends naturally to semi-standard tableaux.
EDIT: On further reflection, the crystal reflection operators FIX all standard Young tableaux (but act nicely on SSYT). So this action is quite boring considered only on SYT.