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The solution of Shareshian-Wachs conjecture by Brosnan-Chow linked togetherthe cohomology of regular semisimple Hessenberg varieties and graded chromaticsymmetric functions on unit interval graphs. On the other hand, it is knownthat unicellular LLT polynomials have similar properties to graded chromaticsymmetric functions. In this paper, we link together the unicellular LLTpolynomials and twin of regular semisimple Hessenberg varieties introduced byAyzenberg-Buchstaber. We prove their palindromicity from topological viewpoint.We also show that modules of a symmetric group generated by faces of apermutohedron are related to a shifted unicellular LLT polynomial and observethe $e$-positivity of shifted unicellular LLT polynomials, which is establishedby Alexandersson-Sulzgruber in general, for path graphs and complete graphsthrough the cohomology of the twins.

Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties