Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties

TL;DR

This paper explores the connections between unicellular LLT polynomials and twin regular semisimple Hessenberg varieties, establishing palindromicity and e-positivity results through topological and cohomological methods.

Contribution

It links unicellular LLT polynomials with twin Hessenberg varieties and proves their palindromicity and e-positivity in specific cases, extending previous conjectures.

Findings

Proved palindromicity of unicellular LLT polynomials from a topological perspective.

Established a relation between modules generated by permutohedron faces and shifted unicellular LLT polynomials.

Observed e-positivity of shifted unicellular LLT polynomials for path and complete graphs.

Abstract

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