Posh Parking Spaces

Abstract

Let $W$ be an irreducible complex reflection group with reflection representation $V$. A $W$-stable, faithful homogeneous system of parameters $\Theta \subseteq \mathrm{Sym}(V^{\ast })$ of common positive degree $p$ is called a posh hsop; $\Theta$ carries a $W$-representation $U$ if $\Theta \simeq U$ as ungraded $W$-modules. We classify the 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 $S/(\Theta )$ is a permutation module for $W$ if and only if $U\simeq V^{\ast }$.

arXiv comment29 pages, 6 tables