Browsing by Author "Ingalls, Colin"

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    A McKay Correspondence for Reflection Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Buchweitz, Ragnar-Olaf; Faber, Eleonore; Ingalls, Colin
    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 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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    The Magic Square of Reflections and Rotations
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Buchweitz, Ragnar-Olaf; Faber, Eleonore; Ingalls, Colin
    We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in 0(3). We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.