Divisors on regular Hessenberg varieties for h=(2,3,4,...,n,n) associated to lower subsets

For a regular Hessenberg variety associated to the Hessenberg function $h=(2,3,4,\dots ,n,n)$, we construct Cartier divisors $D_A$ associated to lower subsets $\mathrm{\varnothing }⊊A⊊[n]$. We show that their first Chern classes satisfy the quadratic relations suggested by the combination of the works of B$\breve{\text{a}}$libanu-Crooks and Horiguchi-Masuda-Shareshian-Song. We also give a characterization of these divisors in terms of the centralizer action. It also follows that these divisors generate the (pseudo)effective cone of divisors on the Hessenberg variety.