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

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 $F_{i,j}$.

💡 Research Summary

This paper presents a profound mathematical bridge between two distinct types of Hessenberg varieties: regular semisimple Hessenberg varieties and regular nilpotent Hessenberg varieties. The research focuses on establishing a structural connection between the coordinate rings of the former and the cohomology rings of the latter, utilizing the concept of polynomial quantization.

The study builds upon the foundational work of Abe, Harada, Horiguchi, and Masuda, who introduced a set of polynomials denoted as $f_{i,j}$. These polynomials are instrumental in providing an explicit presentation of the cohomology rings of regular nilpotent Hessenberg varieties through a clear set of generators and relations. While the $f_{i,j}$ polynomials describe the topological essence of the nilpotent varieties, the paper seeks to expand this framework into the realm of semisimple varieties.

The core methodology employed in this paper is the quantization of the $f_{i,j}$ polynomials, following the sophisticated technique developed by Fomin, Gelfand, and Postnikov. By applying this quantization process, the authors transform the commutative $f_{i,j}$ polynomials into non-commutative counterparts, referred to as $F_{i,j}$. This transition from commutative to non-commutative algebra is a pivotal step in uncovering deeper geometric symmetries that are not visible in the classical setting.

The primary achievement of this research is the proof that these quantized polynomials, $F_{i,j}$, are intrinsically related to the coordinate rings of regular semisimple Hessenberg varieties. This result is highly significant because it establishes a formal link between two seemingly disparate algebraic structures: the coordinate ring (which characterizes the algebraic functions on a variety) and the cohomology ring (which captures the topological properties of a variety). By demonstrating that the quantized version of the nilpotent-related polynomials describes the semisimple-related coordinate rings, the authors provide a unified perspective on the algebraic geometry of Hessenberg varieties.

Furthermore, the paper contributes a quantized recursive formula for $F_{i,j}$. This formula is not merely a theoretical addition but a practical computational tool that allows for the systematic derivation of these complex polynomials. This enables researchers to perform explicit calculations within the coordinate rings of semisimple varieties, facilitating further investigations into their algebraic and combinatorial properties.

In conclusion, this paper successfully bridges the gap between the semisimple and nilpotent regimes of Hessenberg varieties. Through the lens of quantization, it reveals that the algebraic structures of these two types of varieties are deeply interconnected, offering a new, unified framework for studying the intersection of Lie theory, algebraic geometry, and non-commutative algebra.