高维杨图的Schur函数和仿射Yangian

作为统计模型和弦理论中的对称性，仿射Yangian吸引了许多专家的注意。它的表示可以用三维杨图描述，三维杨图的截面是两维杨图，两维杨图的Schur函数、顶点算子实现、玻色-费米对应等结构在表示论、可积系统等领域中有重要的应用，但三维杨图上这些结构还只见雏形。在本项目中，我们计划构造三维杨图上的这些结构，推广到高维并做范畴化。我们采用的方法是把仿射Yangian中三个和为零的参数引入到三维杨图的Schur函数中。利用这种方法，我们已经得到格子数小于等于3的三维杨图的Schur函数，结果发表在《Nuclear Physics B》上。从这些结果来看，此方法是可行的，且我们得到的Schur函数具有关于三个坐标轴对称等性质。本项目将为研究与仿射Yangian有关的不变量和对称性提供新的方法，推动建立三维杨图上与两维杨图平行的结论，且两维结论应该是三维的特殊情况。

As the symmetry in statistical models and string theory, affine Yangian attracts a lot. Its representation can be described by 3D Young diagrams. The slices of 3D Young diagrams are 2D Young diagrams. The structures, such as Schur functions, vertex operator realization, Boson-Fermion correspondence, on 2D Young diagrams are important in representation theory and integrable systems. However, 3D Young diagrams have few of these structures. In this project, we plan to construct these structures, generalize them to the higher dimensional case and do categorification. Our method is to introduce the three parameters of affine Yangian, whose sum is zero, into Schur functions of 3D Young diagrams. By this method, we have obtained the Schur functions of 3D Young diagrams whose total box numbers are less than or equal to 3. This result is published in Nuclear Physics B. It shows that this method is practical, and our Schur functions are symmetric about the three coordinate axes. The results in this project will provide a new way to study the invariants and symmetries related to affine Yangian, and promote the establishment of theories on 3D Young diagrams which are parallel to that on 2D Young diagrams, and theories on 2D Young diagrams should be the special cases of that on 3D Young diagrams.