Posh Parking Spaces

Let $W$ be an irreducible complex reflection group with reflection representation $V$. A $W$-stable, faithful homogeneous system of parameters $\mathrm{\Theta }\subseteq \mathrm{S}\mathrm{y}\mathrm{m}(V^{\ast })$ of common positive degree $p$ is called a \emph{posh hsop}; $\mathrm{\Theta }$ \emph{carries} a $W$-representation $U$ if $\mathrm{\Theta }\simeq U$ as ungraded $W$-modules. We classify the \emph{posh} pairs $(p,U)$ for which a posh hsop of degree $p$ carrying $U$ exists. Every such $U$ is a Galois twist of $V^{\ast }$. Extending work of Ito and Okada, we deduce that the quotient $S\mathrm{/}(\mathrm{\Theta })$ is a permutation module for $W$ if and only if $U\simeq V^{\ast }$.