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

TL;DR

This paper demonstrates that for complex reflection groups G(m,p,2), the endomorphism ring of the reduced hyperplane arrangement provides a non-commutative resolution of the discriminant's coordinate ring, extending previous results.

Contribution

It constructs explicit non-commutative resolutions for G(m,p,2) and decomposes associated matrix factorizations using irreducible representations.

Findings

Endomorphism ring of A(G) is a non-commutative resolution for the discriminant.

Constructs a matrix factorization for the discriminant from A(G).

Provides a full decomposition of the matrix factorization for G(m,p,2).

Abstract

We 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 $\Delta$ 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 $\Delta$ 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 a maximal Cohen--Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement $A(G)$ will be a non-commutative resolution.