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The McKay correspondence and noncommutative resolutions of discriminants of reflection groups 1 Faber, Eleonore
Description
Joint work with Ragnar-Olaf Buchweitz and Colin Ingalls. The classical McKay correspondence relates irreducible representations of a finite subgroup $G$ of $SL(2,\mathbb{C})$ to exceptional curves on the minimal resolution of the quotient singularity $\mathbb{C}^2/G$. Maurice Auslander observed an algebraic version of this correspondence: let $G$ be a finite subgroup of $SL(2,K)$ for a field $K$ whose characteristic does not divide the order of $G$. The group acts linearly on the polynomial ring $S=K[x,y]$ and then the so-called skew group algebra $A=G*S$ can be seen as an incarnation of the correspondence. In particular, $A$ is isomorphic to the endomorphism ring of $S$ over the corresponding Kleinian surface singularity. Auslander's isomorphism holds more generally for small finite subgroups $G$ of $GL(n,K)$, that is, $G$ does not contain any (pseudo-)reflections. The goal of this work is to establish a similar result when $G$ in $GL(n,K)$ is a finite group generated by reflections, assuming that the characteristic of $K$ does not divide the order of the group. Therefore we will consider a quotient of the skew group ring $A=S*G$, where $S$ is the polynomial ring in $n$ variables. We show that our construction yields a generalization of Auslander's result, and moreover, a noncommutative resolution of the discriminant of the reflection group $G$. In particular, we obtain a correspondence between the nontrivial irreducible representations of $G$ and certain maximal Cohen--Macaulay modules over the discriminant, and we can identify some of these modules, namely the so-called logarithmic (co-)residues of the discriminant.
Item Metadata
Title |
The McKay correspondence and noncommutative resolutions of discriminants of reflection groups 1
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2019-09-02T10:07
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Description |
Joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.
The classical McKay correspondence relates irreducible representations
of a finite subgroup $G$ of $SL(2,\mathbb{C})$ to exceptional curves on
the minimal resolution of the quotient singularity $\mathbb{C}^2/G$.
Maurice Auslander observed an algebraic version of this correspondence:
let $G$ be a finite subgroup of $SL(2,K)$ for a field $K$ whose
characteristic does not divide the order of $G$. The group acts linearly
on the polynomial ring $S=K[x,y]$ and then the so-called skew group
algebra $A=G*S$ can be seen as an incarnation of the correspondence. In
particular, $A$ is isomorphic to the endomorphism ring of $S$ over the
corresponding Kleinian surface singularity. Auslander's isomorphism
holds more generally for small finite subgroups $G$ of $GL(n,K)$, that
is, $G$ does not contain any (pseudo-)reflections.
The goal of this work is to establish a similar result when $G$ in
$GL(n,K)$ is a finite group generated by reflections, assuming that the
characteristic of $K$ does not divide the order of the group. Therefore
we will consider a quotient of the skew group ring $A=S*G$, where $S$ is
the polynomial ring in $n$ variables. We show that our construction
yields a generalization of Auslander's result, and moreover, a
noncommutative resolution of the discriminant of the reflection group
$G$. In particular, we obtain a correspondence between the nontrivial
irreducible representations of $G$ and certain maximal Cohen--Macaulay
modules over the discriminant, and we can identify some of these
modules, namely the so-called logarithmic (co-)residues of the discriminant.
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Extent |
50.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Leeds
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Series | |
Date Available |
2020-03-01
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0388808
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Researcher
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International