Superspace coinvariants and inverse systems for $G{L}_n({\mathbb{F}}_q)$

Abstract

Let $q$ be a prime power and write $\Omega$ for the bigraded algebra of regular differential forms over ${\mathbb{F}}_q^n$. The general linear group $G{L}_n({\mathbb{F}}_q)$ acts on $\Omega$; write $SI\subseteq \Omega$ for the ideal generated by $G{L}_n({\mathbb{F}}_q)$-invariants with vanishing constant term. The {\em $G{L}_n({\mathbb{F}}_q)$-superspace coinvariant ring} is the quotient $SR:=\Omega /SI$. We calculate the bigraded Hilbert series of $SR$ and give an operator-theoretic characterization of the inverse system $S{I}^{\perp }$. Our results extend to subgroups $G$ of $G{L}_n({\mathbb{F}}_q)$ which contain $S{L}_n({\mathbb{F}}_q)$.

27 pages