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

2023/07/03 by Young‐Hoon Kiem, Kiem, Young-Hoon, Donggun Lee +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2307.01130

openalex publication_date 2023/07/03 · openalex created_date 2023/07/05 · openalex updated_date 2026/07/28

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.

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