[Paper Review] Parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

Eugene Gorsky|arXiv (Cornell University)|Jun 25, 2020

Algebraic structures and combinatorial models45 references4 citations

TL;DR

This paper constructs actions of rational Cherednik algebras and quantized Gieseker algebras on the equivariant homology of parabolic Hilbert schemes associated to plane curve singularities $\{x^m = y^n\}$ for coprime $m,n$, using the Dunkl-Opdam subalgebra. The key contribution is a combinatorial realization of these symmetries via representation theory of the Cherednik algebra at $t=0$ on non-reduced curves and on singular plane curves.

ABSTRACT

In this note we explicitly construct an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\\mathbb{C}^n)$ corresponding to the permutation representation of $S_n$ on the $\\mathbb{C}^{*}$-equivariant homology of parabolic Hilbert schemes of points on the plane curve singularity $\\{x^{m} = y^{n}\\}$ for coprime $m$ and $n$. We use this to construct actions of quantized Gieseker algebras on parabolic Hilbert schemes on the same plane curve singularity, and actions of the Cherednik algebra at $t = 0$ on the equivariant homology of parabolic Hilbert schemes on the non-reduced curve $\\{y^{n} = 0\\}.$ Our main tool is the study of the combinatorial representation theory of the rational Cherednik algebra via the subalgebra generated by Dunkl-Opdam elements.

Motivation & Objective

• To establish a geometric realization of rational Cherednik algebra actions on the $\mathbb{C}^*$-equivariant homology of parabolic Hilbert schemes over the plane curve singularity $\{x^m = y^n\}$ for coprime $m,n$.
• To extend these actions to quantized Gieseker algebras on the same singular curve geometry.
• To construct actions of the Cherednik algebra at $t=0$ on the equivariant homology of parabolic Hilbert schemes over the non-reduced curve $\{y^n = 0\}$.
• To develop a combinatorial framework for rational Cherednik algebra representation theory via the Dunkl-Opdam subalgebra.
• To unify geometric and algebraic structures in the context of Hilbert schemes of singular curves via representation-theoretic tools.

Proposed method

• Utilizes the Dunkl-Opdam subalgebra of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ as the central algebraic tool.
• Constructs explicit actions of the rational Cherednik algebra on the $\mathbb{C}^*$-equivariant homology of parabolic Hilbert schemes via the combinatorics of the Dunkl-Opdam elements.
• Applies the representation theory of the rational Cherednik algebra to geometric objects—parabolic Hilbert schemes—on the singular curve $\{x^m = y^n\}$.
• Extends the construction to quantized Gieseker algebras by leveraging the same algebraic framework and geometric setting.
• Analyzes the case $t=0$ in the Cherednik algebra to define actions on the equivariant homology of parabolic Hilbert schemes over the non-reduced curve $\{y^n = 0\}$.
• Employs the structure of the permutation representation of $S_n$ on $\mathbb{C}^n$ to guide the construction of the algebraic actions.

Experimental results

Research questions

• RQ1 How can the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ be geometrically realized on the $\mathbb{C}^*$-equivariant homology of parabolic Hilbert schemes over $\{x^m = y^n\}$?
• RQ2 What is the role of the Dunkl-Opdam subalgebra in constructing these actions and enabling the transition to quantized Gieseker algebras?
• RQ3 Can the Cherednik algebra at $t=0$ be realized on the equivariant homology of parabolic Hilbert schemes over the non-reduced curve $\{y^n = 0\}$?
• RQ4 How does the combinatorial representation theory of the Cherednik algebra via the Dunkl-Opdam subalgebra facilitate geometric constructions on singular curve singularities?
• RQ5 What algebraic structures emerge when extending these actions from smooth to non-reduced curve settings?

Key findings

• The paper explicitly constructs an action of the rational Cherednik algebra $H_{1,m/n}(S_n,\mathbb{C}^n)$ on the $\mathbb{C}^*$-equivariant homology of parabolic Hilbert schemes of the plane curve singularity $\{x^m = y^n\}$ for coprime $m,n$.
• This construction leads to the definition of actions of quantized Gieseker algebras on the same geometric objects via the same algebraic framework.
• The authors realize actions of the Cherednik algebra at $t=0$ on the equivariant homology of parabolic Hilbert schemes over the non-reduced curve $\{y^n = 0\}$, extending the geometric picture to singular and non-reduced settings.
• The Dunkl-Opdam subalgebra serves as the key algebraic structure enabling the combinatorial and geometric constructions throughout the paper.
• The representation-theoretic framework based on the Dunkl-Opdam elements provides a unifying mechanism for realizing symmetries on Hilbert schemes of singular curves.
• The results establish a bridge between the combinatorics of rational Cherednik algebras and the geometry of Hilbert schemes on singular and non-reduced curves.