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on Tableaux problem 4

The statement

4.[Fom1994a]*Exam. 2.2.4 notes that lattice of Young diagrams with $\leq r$ rows is differential with degree $r$ . Thus the general theory produces an RSK algorithm from $r$ -colored permutations into pairs of such tableaux with suitable weighting. This should be expandable into similar theories as are developed for Young diagrams and shifted Young diagrams. In particular, (1) Can jeu de taquin be defined for these diagrams? (2) Can a Greene invariant be defined for $r$ -colored permutations that groups permutations that produce the same $P$ tableau?

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