Hessenberg varieties are defined as subvarieties of a full flag variety. This class of subvarities contains, as some special cases, Springer varieties, Peterson varieties, and the toric varieties associated with Weyl chambers. In this talk, we will discuss a relation between cohomology rings of regular nilpotent Hessenberg varieties (e.g. Peterson varieties) and
regular semisimple Hessenberg varieties (e.g. the toric varieties associated with Weyl chambers) in terms of representations of a symmetric group. This is a joint work with Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda.