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

TL;DR

This paper establishes a connection between the coordinate rings of regular semisimple Hessenberg varieties and the cohomology rings of regular nilpotent Hessenberg varieties by quantizing certain polynomials.

Contribution

It introduces a quantization of polynomials related to Hessenberg varieties and links their coordinate rings to cohomology rings, providing new algebraic insights.

Findings

Quantized polynomials $F_{i,j}$ relate to coordinate rings of regular semisimple Hessenberg varieties.

Established a connection between coordinate rings and cohomology rings of Hessenberg varieties.

Provided a recursive formula for the quantized polynomials $F_{i,j}$.

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 $F_{i,j}$.