TITLE: A natural Lascoux-Schützenberger involutions on plane partitions
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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