二元和整数矩阵的双晶体
Double crystals of binary and integral matrices
摘要：
我们定义了一组运算，称为晶体运算，用于元素在{0，1}或N中的矩阵。有水平和垂直的晶体操作，从而产生两个交换结构的晶体图上的这些矩阵。他们提供了一个新的视角在许多方面的RSK对应和它的双重，和相关的建设。根据矩阵的半标准表格的直接编码，操作对应于表格上的晶体操作，分别对应于jeu de taquin幻灯片期间发生的单个移动。我们证明了（对偶）RSK对应和Burge对应作为分解而出现：矩阵M可以通过晶体运算变换为与M相关的编码P和Q符号的每个矩阵，并且可以从P和Q重构。这些分解也可以被解释为计算罗宾逊对应，或作为图片的罗宾逊-申斯特对应。从应用晶体操作的特定方式，可以推导出通过生长图的这些分解的计算，以及要使用的局部规则。我们表明，晶体操作留下一个版本的格林的偏序集不变的定义为矩阵不变，因此，对于这样的问题，在设置的矩阵，他们可以采取的地方的基本克努特变换的话。
英文摘要：
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.
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