Celebratio Mathematica — Benkart

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G. Benkart, F. Sottile, and J. Stroomer: “Tableau switching: Algorithms and applications,” J. Combin. Theory Ser. A 76 : 1 (October 1996), pp. 11–43. MR 1405988 Zbl 0858.05099 article

We define and characterize switching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood–Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schützenberger’s evacuation procedure to switching and in the process obtain further results concerning evacuation. We define reversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood–Richardson coefficients.

Jeffrey D. Stroomer	Related
Frank Sottile	Related