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. $\endgroup$– darij grinbergCommented Nov 10, 2016 at 16:48
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4$\begingroup$ You can act on Young tableaux using the crystal reflection operators, which do indeed satisfy the braid group action. $\endgroup$– Per AlexanderssonCommented Nov 10, 2016 at 17:14
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1$\begingroup$ @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? $\endgroup$– darij grinbergCommented Nov 10, 2016 at 17:43
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$\begingroup$ One reference for the crystal action is Section 1.8 of Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci $\endgroup$– Michael JoyceCommented Nov 10, 2016 at 18:07
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2 Answers
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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.
answered Nov 10, 2016 at 17:14
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1$\begingroup$ I'd rather go with Shimozono's exposition for readability, though :) $\endgroup$ Commented Nov 10, 2016 at 17:44
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$\begingroup$ 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 ? $\endgroup$– jackCommented Nov 14, 2016 at 2:46
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$\begingroup$ This action also permutes the weight, so perhaps this action is not that interesting on SYTs, but only on SSYTs... $\endgroup$ Commented 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.
answered Nov 10, 2016 at 17:23
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3$\begingroup$ "On further reflection...": nice pun :) $\endgroup$ Commented Apr 9, 2021 at 18:59