Browsing by Author "Faber, Eleonore"

Now showing 1 - 4 of 4
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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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    Resolutions in Local Algebra and Singularity Theory
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Conca, Aldo; Cutkosky, Steven Dale; Faber, Eleonore; Iyengar, Srikanth B.
    Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world "resolution" is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.
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    Swallowtail on the shore
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Buchweitz, Ragnar-Olaf; Faber, Eleonore
    Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetries, move on to groups, singularities, and finally find the connection between a tetrahedron and a “swallowtail”. Our running example is the tetrahedron, but every construction can be carried out with any other of the Platonic solids.
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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.