Bijective Combinatorics of Young Tableaux

在过去的几年里，在计数代数几何，特别是Gromov-Witten理论和代数组合学，特别是Young Tableaux恒等式之间，有越来越多的联系。作者打算继续研究与这些恒等式相关的组合双射。特别是，他打算研究某些经典组合恒等式的多重加权类似，这些恒等式也自然地产生于对称群的表示理论。他还打算探索离散随机过程的计数、代数和概率应用，以生成加权Young表。Young表是一百多年前在研究一大类具有对称性的空间时引入的基本组合结构。多年来，人们结合统计物理、离散概率以及最近的代数几何和场论等其他研究领域对Young Tableaux进行了进一步的研究。审查员打算追求后一种联系的组合方面，其内涵是开发新的工具，这反过来将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.