Bijective Combinatorics of Young Tableaux

在过去的几年里，数列代数几何（特别是Gromov-Witten理论）和代数组合学（特别是Young表恒等式）之间的联系越来越多。提议者打算继续研究与这些恒等式相关的组合双反。特别地，他打算研究某些经典组合恒等式的多重加权类似物，这些恒等式也是从对称群的表示理论中自然产生的。他还打算研究离散随机过程在加权Young表生成中的枚举、代数和概率应用。年轻的画面是基本的组合结构，它是在一百多年前引入的，与研究一类具有对称性的空间有关。多年来，Young tableaux与其他研究领域的联系得到了进一步的研究，如统计物理学、离散概率，以及最近的代数几何和场论。审稿人打算研究后一种联系的组合方面，其意图是开发新的工具，从而在这两个领域产生新的结果。

In the past few years there has been a growing number of connections between enumerative algebraic geometry, in particularly Gromov-Witten theory, and the algebraic combinatorics, in particular Young tableaux identities. The proposer intend to pursue the study of combinatorial bijections related to these identities. In particular, he intends to investigate multi-weighted analogues of certain classical combinatorial identities which also arise naturally from the representation theory of the symmetric group. He also intends to pursue both enumerative, algebraic and probabilistic applications of the discrete random processes for generation of weighted Young tableaux.Young tableaux are basic combinatorial structure which were introduced over a hundred years ago in connection with studies of a large class of spaces with symmetries. Over the years, Young tableaux have been further studied in connection with other fields of studies, such as statistical physics, discrete probability, and more recently, algebraic geometry and field theory. The reviewer intends to pursue the combinatorial aspects of the latter connections, with the intension of developing new tools, which in turn will lead to new results in both fields.