Abstract
AbstractWe show that for the family of complex reflection groups G = G(m, p,2) appearing in the Shephard–Todd classification, the endomorphism ring of the reduced hyperplane arrangement A(G) is a non-commutative resolution for the coordinate ring of the discriminant Δ of G. This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for Δ from A(G) and decompose it using data from the irreducible representations of G. For G(m, p,2) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen–Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement A(G) will be a non-commutative resolution. For the groups G(m,1,2), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
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