Eunjeong Lee, Geometry of Flag Varieties and Related Combinatorics

For a semisimple algebraic group G and a Borel subgroup B, the homogeneous space G/B, called the flag variety, is a smooth projective variety which has a fruitful connection with G-representations. Indeed, the set of global sections H⁰(G/B, L) is an irreducible G-representation for a very ample line bundle L on G/B. On the other hand, string polytopes are combinatorial objects which encode the characters of irreducible G-representations. One of the most famous examples of string polytopes is the Gelfand–Cetlin polytope, and there might exist combinatorially different string polytopes. The string polytopes are related to the flag varieties via the theory of Newton–Okounkov bodies. In this talk, we will study Gelfand–Cetlin type string polytopes, their enumerations, and we will present small toric resolutions of certain string polytopes.  This talk is based on several collaborations with Yunhyung Cho, Jang Soo Kim, Yoosik Kim, and Kyeong-Dong Park.