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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.

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  • $\begingroup$ 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. $\endgroup$ Nov 10, 2016 at 16:48
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    $\begingroup$ You can act on Young tableaux using the crystal reflection operators, which do indeed satisfy the braid group action. $\endgroup$ Nov 10, 2016 at 17:14
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    $\begingroup$ @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? $\endgroup$ Nov 10, 2016 at 17:43
  • $\begingroup$ One reference for the crystal action is Section 1.8 of Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci $\endgroup$ Nov 10, 2016 at 18:07

2 Answers 2

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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.

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    $\begingroup$ I'd rather go with Shimozono's exposition for readability, though :) $\endgroup$ Nov 10, 2016 at 17:44
  • $\begingroup$ 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 ? $\endgroup$
    – jack
    Nov 14, 2016 at 2:46
  • $\begingroup$ This action also permutes the weight, so perhaps this action is not that interesting on SYTs, but only on SSYTs... $\endgroup$ Apr 9, 2021 at 20:29
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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.

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    $\begingroup$ "On further reflection...": nice pun :) $\endgroup$ Apr 9, 2021 at 18:59

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