  The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet
tied the graded chromatic symmetric functions on indifference graphs (or unit
interval graphs) and the cohomology of regular semisimple Hessenberg varieties
with the dot action. A similar result holds between unicellular LLT polynomials
and twins of regular semisimple Hessenberg varieties. A recent result by
Abreu-Nigro enabled us to prove these results by showing the modular law for
the geometrical objects, and this is indeed done by Precup-Sommers and
Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular
law through GKM theory.
