# Symmetric group action on Young Tableaux

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 would be quite interesting to have. I don't know any (except for the trivial one). The Bender-Knuth involutions fail to satisfy the braid relations (even for standard tableaux). The Young seminormal form is an action on a vector space, not on a set. Nov 10, 2016 at 16:48
• You can act on Young tableaux using the crystal reflection operators, which do indeed satisfy the braid group action. Nov 10, 2016 at 17:14
• @PerAlexandersson: Nice! I'm wondering how I forgot that. And that's a transitive group action, so it even explains why the number of standard Young tableaux of a given shape divides $n!$. Maybe it can be strengthened to a proof of the hook-length formula? Nov 10, 2016 at 17:43
• One reference for the crystal action is Section 1.8 of Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci Nov 10, 2016 at 18:07

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.

• I'd rather go with Shimozono's exposition for readability, though :) Nov 10, 2016 at 17:44
• There is a natural action of the Weyl group (In this case its Sn) on the zero weight space that is the $T$ invariants of the Global sections of of the line bundle $L_{\lambda$ on $G/B$ which has a basis of standard tableaux with some restriction on entries. What is this action and how does it related to any of the above actions ?
– jack
Nov 14, 2016 at 2:46
• This action also permutes the weight, so perhaps this action is not that interesting on SYTs, but only on SSYTs... Apr 9, 2021 at 20:29

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.

• "On further reflection...": nice pun :) Apr 9, 2021 at 18:59