Permutation module decomposition of the cohomology of Hessenberg varieties associated with lollipop graphs

We study the cohomology of regular semisimple Hessenberg varieties associated with lollipop graphs as a module under the dot action. Using the natural basis introduced by Cho, Hong, and Lee, which we call the CHL basis, we establish structural properties of the dot action, including a result for classes satisfying $i$-decomposability. We also obtain an explicit elementary symmetric function expansion of the chromatic quasisymmetric functions of lollipop graphs in terms of $h$-admissible permutations and their associated partitions. Combining these geometric and combinatorial results, we construct a permutation module decomposition of the cohomology of the corresponding Hessenberg varieties, thereby proving a conjecture of Cho, Hong, and Lee for lollipop graphs.