Modular Invariants of Finite Affine Linear Groups

Abstract

We study modular invariants of finite affine linear groups over a finite field $\mathbb {F}_{q}$ under affine actions and linear actions. We generalize a result of Chuai (J Algebra 318:710–722, 2007, Theorem 4.2) to any m-folds affine actions. Suppose $G\leqslant \mathrm{GL}(n,\mathbb {F}_{q})$ is a subgroup and W denotes the canonical module of $\mathrm{GL}(n,\mathbb {F}_{q})$. We denote by $\mathbb {F}_{q}[W]^{G}$ the invariant ring of G acting linearly on W and denote by $\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}$ the invariant ring of the affine group $AG(W^{*})$ of G acting canonically on $W_{n+1}:=W\oplus \mathbb {F}_{q}$. We show that if $\mathbb {F}_{q}[W]^{G}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s}]$, then $\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s},h_{n+1}]$, where $h_{n+1}$ denotes the $(n+1)$-th Mui’s invariant of degree $q^{n}$. Let $\mathrm{AGL}_{1}(\mathbb {F}_{p})$ be the 1-dimensional affine general linear groups over the prime field $\mathbb {F}_{p}$. We find a generating set for the ring of vector invariants $\mathbb {F}_{p}[mW_{2}]^{\mathrm{AGL}_{1}(\mathbb {F}_{p})}$ and determine the Noether’s number $\upbeta _{mW_{2}}(\mathrm{AGL}_{1}(\mathbb {F}_{p}))$ for any $m\in \mathbb {N}^{+}$.