Some Bijective Correspondences Involving Domino Tableaux

Source abstract

We define a number of new combinatorial operations on skew semistandard domino tableaux that complement constructions defined by C. Carré and B. Leclerc, and clarify the link with ordinary skew semistandard tableaux and 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, and which determine the "plactic class" of the domino tableau; (2) a weight preserving reversible transformation of domino tableaux into ordinary tableaux of a related shape (the correspondence involves 2-quotients) mapping the subset of Yamanouchi domino tableaux onto that of the 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 horizontally and vertically by a factor $2$. The essential properties of (1) and (2) are obtained by proving their commutation with the "coplactic" (or crystal) operations (which for domino tableaux were defined by Carré and Leclerc). Construction (2) allows algorithmic separation of the Littlewood-Richardson tableaux describing the decomposition of the tensor square of a general linear group representation into contributions to its symmetric and alternating parts.