Coordinate rings of regular semisimple Hessenberg varieties and cohomology rings of regular nilpotent Hessenberg varieties

2025/12/12 by Horiguchi, Tatsuya
#05E05 #14M15 #14N35 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.11565

Abstract

The polynomials f_i,j are introduced by Abe-Harada-Horiguchi-Masuda to produce an explicit presentation by generators and relations of the cohomology rings of regular nilpotent Hessenberg varieties. In this paper we quantize the polynomials f_i,j by a method of Fomin-Gelfand-Postnikov. Our main result states that their quantizations F_i,j are related to the coordinate rings of regular semisimple Hessenberg varieties. This result yields a connection between the coordinate rings of regular semisimple Hessenberg varieties and the cohomology rings of regular nilpotent Hessenberg varieties. We also provide the quantized recursive formula for F_i,j.

Citations

• On singularity and normality of regular nilpotent Hessenberg varieties
• The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials
• Hessenberg varieties and hyperplane arrangements
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties
• Chromatic quasisymmetric functions
• Exponents for B-stable ideals
• Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties
• Quantum cohomology of flag varieties

Related

• The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials
• A filtration on the cohomology rings of regular nilpotent Hessenberg varieties
• The cohomology rings of regular nilpotent Hessenberg varieties
• Regular nilpotent partial Hessenberg varieties
• Higher Specht bases and q-series for the cohomology rings of certain Hessenberg varieties