Geometry of the twin manifolds of regular semisimple Hessenberg varieties and unicellular LLT polynomials

TL;DR

This paper explores the geometry of twin manifolds associated with regular semisimple Hessenberg varieties, providing a geometric proof of the LLT Shareshian-Wachs conjecture linking cohomology to unicellular LLT polynomials.

Contribution

It establishes explicit geometric relations between twin manifolds and offers a direct cohomological proof of the LLT Shareshian-Wachs conjecture.

Findings

Twin manifolds are related by blowups and fiber bundles.

Cohomology of twin manifolds confirms the LLT Shareshian-Wachs conjecture.

Provides geometric insight into the structure of Hessenberg varieties.

Abstract

Recently, Masuda-Sato and Precup-Sommers independently proved an LLT version of the Shareshian-Wachs conjecture which says that the Frobenius characteristics of the cohomology of the twin manifolds of regular semisimple Hessenberg varieties are unicellular LLT polynomials. The purpose of this paper is to study the geometry of twin manifolds and we prove that they are related by explicit blowups and fiber bundle maps. Upon taking their cohomology, we obtain a direct proof of the modular law which establishes the LLT Shareshian-Wachs conjecture.