A McKay correspondence for reflection groups

Abstract

We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A = S * G. If G is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring S G /(∆) 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 S G /(∆). 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 S G /(∆). We identify some of the corresponding matrix factorizations, namely the so-called logarithmic co-residues of the discriminant.In order to show that A is an endomorphism ring, we first view A as a CM-module over the (noncommutative) ring T * H and will use the functor i * : Mod(T * H) − → Mod(R/(∆)) , coming from a standard recollement. For this part we will need that G is a true reflection group, that is, generated by reflections of order 2. Then clearly H ∼ = µ 2 . In order to use the 1 Most of our results also hold if the characteristic of the field K does not divide the order |G| of the group G. However, in order to facilitate the presentation, we restrict to K = C.which gives us that the map S J − → S decomposes into components of the form Hom KG (U, S) ⊗ K U J − → Hom KG (U ⊗ det, S) ⊗ K U ⊗ det for each irreducible representation U of G.

McKay Quivers and Lusztig Algebras of Some Finite Groups

“…We are interested in complex reflection groups, since in [16] the first three authors established a McKay correspondence for true reflection groups, i.e., groups generated by complex reflections of order 2: the nontrivial irreducible representations of such a group G were shown to correspond to maximal Cohen-Macaulay modules over the discriminant of the reflection group, in particular, they correspond to the isotypical components of the hyperplane arrangement, viewed as a module over the coordinate ring of the discriminant, see [16,Thm. 4.17].…”

“…Finally a comment on how this paper came into being: In the first preprint version on the arXiv of [16] we had already determined the McKay quivers of G(1, 1, n), and Section 3 of the current paper appeared originally there. However, it made sense to combine it with the results of M. Lewis' thesis about the McKay quivers of the general G(r, p, n) [36], that now make up Sections 4 and 5 of the current paper.…”

“…This is in particular in the literature on representation theory, see e.g. [3,4,8,35,53], because (G) op for G ⊆ GL(2, K) then coincides with the Auslander-Reiten quiver of G. Our convention is used in the geometric context, see [16,21,25].…”

Non-commutative resolutions for the discriminant of the complex reflection group $G(m,p,2)$

“…Theorem 1.1 is an extension of [BFI20] where, for a complex reflection group generated by reflections of order 2, it is shown that End R/∆ S/(z) is a NCR for R/(∆). The groups G(2k, k, 2) are generated by order 2 reflections.…”

Noncommutative resolutions of discriminants

“…On the other hand, we give a short introduction to discriminants of finite reflection groups and our construction of their noncommutative desingularizations. The details of our new results will be published elsewhere [BFI16].…”