Modular Invariants of Finite Affine Linear Groups

Abstract

We study modular invariants of finite affine linear groups over a finite field Fq 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⩽ GL (n, Fq) is a subgroup and W denotes the canonical module of GL (n, Fq). We denote by Fq[ W] G the invariant ring of G acting linearly on W and denote by Fq[Wn+1]AG(W∗) the invariant ring of the affine group AG(W∗) of G acting canonically on Wn+1: = W⊕ Fq. We show that if Fq[ W] G= Fq[ f1, f2, … , fs] , then Fq[Wn+1]AG(W∗)=Fq[f1,f2,…,fs,hn+1], where hn+1 denotes the (n+ 1) -th Mui’s invariant of degree qn. Let AGL 1(Fp) be the 1-dimensional affine general linear groups over the prime field Fp. We find a generating set for the ring of vector invariants Fp[mW2]AGL1(Fp) and determine the Noether’s number βmW2(AGL1(Fp)) for any m∈ N+.