Modular law through GKM theory

2023/10/24 by Tatsuya Horiguchi, Horiguchi, Tatsuya, Mikiya Masuda +3
Mathematics · #05A05 #14M15 (Secondary) #57S12 (Primary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.16235

openalex publication_date 2023/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Related

• Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties
• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• Unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties
• Chromatic symmetric functions from the modular law
• The second cohomology of regular semisimple Hessenberg varieties from GKM theory