15 March 2020 A McKay correspondence for reflection groups
Ragnar-Olaf Buchweitz, Eleonore Faber, Colin Ingalls
Duke Math. J. 169(4): 599-669 (15 March 2020). DOI: 10.1215/00127094-2019-0069


We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A=SG. If G is generated by order 2 reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring SG/(Δ) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen–Macaulay modules over the coordinate ring SG/(Δ). These maximal Cohen–Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over SG/(Δ). We identify some of the corresponding matrix factorizations, namely, the so-called logarithmic (co-)residues of the discriminant.


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Ragnar-Olaf Buchweitz. Eleonore Faber. Colin Ingalls. "A McKay correspondence for reflection groups." Duke Math. J. 169 (4) 599 - 669, 15 March 2020. https://doi.org/10.1215/00127094-2019-0069


Received: 20 July 2018; Revised: 31 July 2019; Published: 15 March 2020
First available in Project Euclid: 20 February 2020

zbMATH: 07198463
MathSciNet: MR4072636
Digital Object Identifier: 10.1215/00127094-2019-0069

Primary: 14E16
Secondary: 13C14 , 14A22 , 14E15

Keywords: hyperplane arrangements , matrix factorizations , maximal Cohen–Macaulay modules , noncommutative desingularization , reflection groups

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 4 • 15 March 2020
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