Tableau Switching: Algorithms and Applications

Abstract

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 Schu tzenberger'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.

A simplified Kronecker rule for one hook shape

“…Suppose < and ≺ are total orders of A that are identical except for the order of a and b, say a < b but b ≺ a. Then there is a natural bijection between colored tableaux with respect to < and colored tableaux with respect to ≺ via a process called conversion, introduced by Haiman [6] (see also [2]). …”

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

“…We will need the following fact (the "infusion involution"); compare [Haiman 1992;Benkart et al 1996]. …”

Reductions of Young Tableau Bijections

“…The reversal of a skew tableau can be characterized by means of dual equivalence [16], although this is not the case for Schützenberger involution in the case of skew shapes. Proposition 15 is proved in §5 of [6] and follows from the fact that it is defined as the composition of three bijections and relations ( ). Bender-Knuth transformations were introduced by E. Bender and D.E.…”

“…Proposition Tableau switching, defined by means of the Bender-Knuth transformations, was used by James and Kerber [21, §2.8] in a proof of the LR-rule. More recently, the tableau switching map was defined in a more general context in [6]; there its main properties were established. This is what we call the Tableau Switching map.…”

“…See also [27, §2] for applications of tableau switching. Proposition 2 is Proposition 2.6 in [6], Proposition 4 is contained in [6, §2], and Proposition 16 is a part of Theorem 3.1 in [6].…”