Double crystals of binary and integral matrices

Abstract

We define a set of operations called crystal operations on matrices with entries either in {0,1} or in N. There are horizontal and vertical crystal operations, giving rise to two commuting structures of a crystal graph on these matrices. They provide a new perspective on many aspects of the RSK correspondence and its dual, and related constructions. Under a straightforward encoding of semistandard tableaux by matrices, the operations correspond to crystal operations on tableaux, respectively to individual moves occurring during a jeu de taquin slide. We show that the (dual) RSK correspondence and the Burge correspondence arise as decompositions: a matrix M can be transformed by crystal operations into each of the matrices encoding P and Q symbol associated to M under these correspondences, and it can be reconstructed from P and Q. These decomposition can also be interpreted as computing Robinson's correspondence, or as the Robinson-Schensted correspondence for pictures. From a particular way of applying crystal operations, the computation of these decompositions by growth diagrams can be deduced, as well as the local rules that are to be used. We show that that crystal operations leave a version of Greene's poset invariant defined for matrices unchanged, so that for such questions in the setting of matrices they can take the place of elementary Knuth transformations on words.