Some bijective correspondences involving domino tableaux

Abstract

We define a number of new combinatorial operations on skew semistandard domino tableaux, which together with constructions introduced earlier by C. Carre and B. Leclerc, define an elegant structure on the set of these tableaux, that closely parallels the structure defined in [math.CO/9908099] for ordinary skew semistandard tableaux, that is related to the Littlewood-Richardson rule. These operations are: (1) a bijection between semistandard domino tableaux and certain pairs of ordinary tableaux of the same weight that together fill the same shape; (2) a weight preserving transformation of domino tableaux to ordinary tableaux of a related shape (the correspondence involves 2-quotients) with the property that Yamanouchi domino tableaux correspond precisely to Littlewood-Richardson tableaux; (3) a correspondence between Yamanouchi domino tableaux of shape $\lambda$ and weight $\mu$ and Yamanouchi domino tableaux of shape $\mu'$ and weight $\lambda$, where $\mu'$ is $\mu$ scaled by a factor 2. The property at the heart of the structure so defined is that operations (1) and (2) commute with the coplactic operations (implicitly) defined by Carre and Leclerc. (1) elucidates their bijection analogous to that of Robinson (a.k.a. the Robinson-Schensted correspondence), while the operation of moving chains in domino tableaux underlying (2) is very similar to jeu de taquin.