The combinatorics of alternating sign matrices and plane partitions

一个激动人心的新研究项目，其目的是利用涉及偏序集的创新技术，大幅提高对交替符号阵列和广义平面划分的内在组合性质的理解。交替符号矩阵是由0、1和-1项组成的正方形数组，其中非零项沿每行和列交替符号，以1开始和结束。平面划分是具有非负整数项的矩形数组，每行从左到右，从上到下每列递减。虽然这些物体有完全初级的定义，但它们也具有复杂而神秘的组合特征，这些特征的研究需要应用一系列高级数学领域的复杂技术。例如，尽管交替符号矩阵和平面划分表面上看起来不同，但它们通过深层计数联系在一起。具体地说，在某些自然有限集中，这些对象的数量之间存在等式，虽然这些等式可以非常简单地表述，但目前它们只能用复杂和高度技术性的方法来验证。本项目的主要目标是简化和阐明这些问题，实现这一目的的方法论包括以某些自然方式扩大交替符号矩阵和平面划分的类别，然后应用各种新的技术。具体地说，将考虑具有关于其形状和条目的更一般条件的数组，并将使用最近开发的来自动态代数组合学的强大工具，涉及对偏序集合的作用。

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.