We discuss the representations of the symmetric groups on the cohomology of two types of moduli spaces: (1) the moduli of pointed rational curves and (2) regular semisimple Hessenberg varieties.
The moduli space of pointed rational curves has a natural action of the symmetric group which permutes the marked points. In the first half, we present a formula for the characters of the induced representations on its cohomology. This reduces the computation to a purely combinatorial problem. As a corollary, we show that the following representations are permutation representations: (i) the representation on the k-th cohomology with k at most 6 and (ii) the sum of two cohomology groups of subsequent even degrees. A key ingredient is wall crossings of delta-stability on moduli of quasi-maps. This is a joint work with Prof. Jinwon Choi and Young-Hoon Kiem.
Hessenberg varieties are interesting objects in both algebro-geometric and combinatorial perspectives. There is a naturally defined action of the symmetric group on their cohomology. The Shareshian-Wachs conjecture connects their characters with the chromatic quasi-symmetric functions of the associated graphs, which are certain refinements of the chromatic polynomials of graphs. In the second half, we study the birational geometry of Hessenberg varieties via blowups by introducing generalized Hessenberg varieties. Based on explicit geometry, we obtain two new recursive algorithms for the characters. Moreover, we also provide an elementary proof of the Shareshian-Wachs conjecture and its natural generalization. This is a joint work with Prof. Young-Hoon Kiem.