## Abstract

We show how to compute a Macaulay dual class in local cohomology $\phantom{\rule{-0.56905pt}{0ex}}{H}_{\U0001d52a}^{n}\left(\mathbb{F}\left[V\right]\right)\phantom{\rule{-0.85358pt}{0ex}}$ for an given ideal $I\left(\mathcal{S}\right)\subset \mathbb{F}\left[V\right]$ of generalized invariants associated to a set of reflections $\mathcal{S}\subset \text{GL}(n,\mathbb{F})$ that generate a finite subgroup $G\left(\mathcal{S}\right)\le \text{GL}(n,\mathbb{F})$. We then apply this in the case that the coinvariant algebra $\mathbb{F}{\left[V\right]}_{G\left(\mathcal{S}\right)}$ is a fixed point free Poincaré duality algebra to obtain a Macaulay dual for the Hilbert ideal $\U0001d525\left(G\right(\mathcal{S}\left)\right)$ of $G\left(\mathcal{S}\right)$.

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