Rational Cherednik algebras and Hilbert schemes
type: Item
instance of: scholarly article
title: Rational Cherednik algebras and Hilbert schemes
zbMATH Open document ID: 1084.14005
author: J. T. Stafford
author: Iain Gordon
published in: Advances in Mathematics
publication date: 2005-12-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0407516
full work available at URL: Rational.pdf
review text: The article strengthens the connections between Cherednik algebras and geometry by showing that they can be regarded as noncommutative deformations of Hilbert schemes of points in the plane. The article explicitly defines the rational Cherednik algebra \(H_c\) of type \(A_{n-1}\), and its spherical subalgebra \(U_c\). Let \(W=\mathfrak S_n\) be the symmetry group on \(n\) letters, regarded as the Weyl group of type \(A_{n-1}\) acting on its \((n-1)\)-dimensional representation \(\mathfrak h\in\mathbb{C}^n\) by permutations. Then \(H_c\) may be regarded as a deformation of the twisted group ring \(D(\mathfrak h)\ast W\), where \(D(\mathfrak h)\) is the ring of differential operators on \(\mathfrak h \) with the natural action of the symmetric group \(W=\mathfrak S_n.\) The algebra \(U_c\) is then the corresponding deformation of the fixed ring \(D(\mathfrak h)^W.\) \(U_c\) and \(H_c\) both have a natural filtration by order of differential operators, with associated graded rings \(\operatorname{gr}U_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W\) and \(\operatorname{gr}H_c\cong\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]\ast W\), and so \(U_c\) may be regarded as a deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W.\) The article is mostly concerned about \(U_c\), but the authors prove that \(U_c\) and \(H_c\) are Morita equivalent. It is known that the map \(\tau:\text{Hilb}^n\mathbb{C}\rightarrow \mathbb{C}^{2n}/W\) is a resolution of singularities, and Haiman has described \(\text{Hilb}^n\) as ``Proj'' of Rees rings. This article proves that for \(\text{Hilb}(n)=\tau^{-1}(\mathfrak h\oplus\mathfrak h^\ast/W)\), \(\tau:\text{Hilb}(n)\rightarrow \mathfrak h\oplus\mathfrak h^\ast/W \) is a crepant resolution of singularities. The ring \(U_c\) has finite global homological dimension so one should expect the properties of a smooth deformation of \(\mathbb{C}[\mathfrak h\oplus\mathfrak h^\ast]^W,\) that is, its properties should be more closely re…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 14A22
zbMATH DE Number: 2240054
zbMATH Keywords: \(\mathbb Z\)-algebras
zbMATH Keywords: resolution of singularities
MaRDI profile type: Publication
OpenAlex ID: W2130905271
arXiv ID: math/0407516
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DOI: 10.1016/J.AIM.2004.12.005
description: scientific article