# Matrix Factorisations Arising From Well-Generated Complex Reflection Groups

## TL;DR

This paper explores a duality in complex reflection groups, providing a representation-theoretic realization, and constructs matrix factorizations to analyze invariants and differentials, with applications to geometry and algebra.

## Contribution

It offers a new representation-theoretic approach to duality in complex reflection groups and constructs explicit matrix factorizations for invariant modules.

## Key findings

- Explicit matrix factorizations of basic invariants
- New formulas for graded dimensions of modules
- Application to cohomology of flag manifolds

## Abstract

We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally identify invariant vector fields with vector fields on the orbit space, for the action of a duality group. As another application, we construct matrix factorisations of the highest degree basic invariant which give free resolutions of the module of K\"{a}hler differentials of the coinvariant algebra $A$ associated to such a reflection group. From this one can explicitly calculate the dimension of each graded piece of $\Omega_{A/\mathbb{C}}$ and of ${\rm Der}_{\mathbb{C}}(A,A)$, adding a new formula to the numerology of reflection groups. This applies for instance when $A$ is the cohomology of any complete flag manifold, and hence has geometric consequences.