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The Lascoux-Schützenberger involutions, $s_i$, that permute the weight of semi-standard Young tableaux are fairly known.

They satisfy some nice Coxeter relations, for example, if $v$ and $w$ are reduced words of the same permutation, then $s_{v_1} \dots s_{v_l} = s_{w_1} \dots s_{w_l}$.

Each $s_i$ only acts on the entries $i$ and $i+1$ in the tableau, and columns that contain one entry of both $i$ and $i+1$ is fixed.

Now, is there some generalization of these involutions that

  • Generalize to plane partitions (or arbitrary fillings), where the same element can appear more than once in both rows and columns,
  • satisfy the independence of reduced word relation
  • specialize to the classical involutions on semi-standard Young tableaux?

Note that this generalization does not need to preserve the position of other entries if the filling is not a SSYT: What I am looking for will not have this property.

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There is a modern viewpoint on combinatorics Young tableaux in Danilov, V. I.; Koshevoy, G. A. Massifs and the combinatorics of Young tableaux. (Russian) Uspekhi Mat. Nauk 60 (2005), no. 2(362), 79--142; translation in Russian Math. Surveys 60 (2005), no. 2, 269–334 (the LS-involutions are considered in Appendix A).

This paper is online here

http://www.mathnet.ru/links/b2797cbfa2d63906937371dbe7b7b519/rm1402.pdf

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  • $\begingroup$ Is it possible to access this article online? I tried to find it, but was unable to... $\endgroup$ – Per Alexandersson Sep 20 '15 at 3:45
  • $\begingroup$ The title of the article is often translated into English as "Arrays and the combinatorics of Young tableaux". I couldn't find it online, but you might look at this paper arxiv.org/abs/math/0504299, which has a very brief (two page) summary of the main notions associated with arrays in it and is based on a similar viewpoint as the aforementioned paper. $\endgroup$ – Michael Joyce Oct 20 '15 at 0:15

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