The combinatorics of alternating sign matrices and plane partitions

一个令人兴奋的新的研究项目，其目的是获得交替符号阵列和广义平面分区的内在组合性质的实质性增强的理解，使用涉及偏序集的创新技术。交替符号矩阵是具有条目0、1和-1的正方形阵列，其中非零条目在符号上沿沿着每一行和每一列交替，以1开始和结束。一个平面分割是一个矩形数组，其中的非负整数元素在每一行上从左到右、在每一列上从上到下逐渐减少。虽然这些对象有完全基本的定义，但它们也具有复杂和神秘的组合特征，其研究需要应用来自一系列高级数学领域的复杂技术。例如，尽管它们表面上不同的外观，交替符号矩阵和平面划分是由深层枚举连接。具体地说，在某些自然有限集合中，这些物体的数量之间存在相等性，尽管这些相等性可以非常简单地陈述，但目前只能使用复杂和高度技术性的方法来验证。通过用于实现这一点的方法包括以某些自然的方式拓宽交替符号矩阵和平面划分的类别，然后应用各种新技术。特别是，阵列与更一般的条件，其形状和条目将被考虑，和最近开发的强大的工具，从动态代数组合，涉及行动的偏序集，将被采用。

an exciting new research project whose aim is to obtain a substantially enhanced understanding of the intrinsic combinatorial properties of alternating sign arrays and generalized plane partitions, using innovative techniques involving partially ordered sets. An alternating sign matrix is a square array with entries 0, 1 and -1, in which the nonzero entries alternate in sign along each row and column, starting and ending with a 1. A plane partition is a rectangular array with nonnegative integer entries which decrease weakly from left to right across each row, and from top to bottom down each column. Although these objects have completely elementary definitions, they also possess intricate and enigmatic combinatorial characteristics whose study has required the application of sophisticated techniques from a range of advanced mathematical areas. For instance, despite their superficially-different appearances, alternating sign matrices and plane partitions are linked by deep enumerative connections. Specifically, there are equalities between the numbers of these objects in certain natural finite sets, and although these equalities can be stated very simply, they can only currently be verified using complicated and highly technical approaches.The primary objective of this project is to simplify and elucidate such matters, with the methodology for achieving this consisting of widening the classes of alternating sign matrices and plane partitions in certain natural ways, and then applying various novel techniques. In particular, arrays with more general conditions on their shapes and entries will be considered, and recently-developed powerful tools from dynamical algebraic combinatorics, involving actions on partially ordered sets, will be employed.