Some matrix factorizations of discriminants of reflection groups
Let G be a finite group in GL(n,C) generated by reflections.
Then G acts on the vectorspace V=C^n, as well as on the polynomial ring in n variables S=Sym_C(V). It is well-known that the reflection arrangement in V and the discriminant D of G in the quotient V/G both have singular locus of codimension 1 and are so-called free divisors.
Matrix factorizations first appeared in work of D. Eisenbud and are a simple way to describe maximal Cohen-Macaulay modules over hypersurface singularities.
In this talk I will present certain matrix factorizations (and hence maximal Cohen-Macaulay modules) that arise from our construction of a noncommutative resolution of singularities of the discriminant D as a quotient of the twisted group ring S*G. In particular, we can interprete the modules of logarithmic derivations and modules of logarithic residues in terms of certain irreducible representations of G. This is joint work with R.-O. Buchweitz and C. Ingalls.