Junhao Fan L&S Sciences
Schubert Calculus Through Toric Geometry
In our research, we will use toric geometry to study the cohomological structure of complex Grassmannians. The cohomology ring of a Grassmannian varieties is described by the Littlewood-Richardson rule. One of the main open questions in Schubert calculus concerns the generalization of the Littlewood-Richardson rule to flag varieties. Such a generalization is highly desirable, because it is a manifestly positive formula that can be applied to other areas: in algebraic geometry, it helps describe complicated intersections; in representation theory, it helps to find irreducible, direct-sum decompositions of tensor products; in physics, it can be applied to calculate certain physical quantities. Our research aims to give a new, geometric proof of the Littlewood-Richardson rule,by applying toric degeneration to Weyl-group-translated Schubert varieties. More specifically, we will study the intersection behavior of Schubert varieties, in terms of face-intersections of Gelfand-Cetlin polytopes. A new geometric perspective would help give a deeper understanding of the Littlewood-Richardson rule, particularly in its relation to Schubert calculus. New methods could also suggest how to generalize the Littlewood-Richardson rule to arbitrary flag varieties.