Quasi-monomial actions and some 4-dimensional rationality problems

Let G   be a finite group acting on k(x1,…,xn)k(x1,…,xn), the rational function field of n variables over a field k  . The action is called a purely monomial action if σ⋅xj=∏1⩽i⩽nxiaij for all σ∈Gσ∈G, for 1⩽j⩽n1⩽j⩽n where (aij)1⩽i,j⩽n∈GLn(Z)(aij)1⩽i,j⩽n∈GLn(Z). The main question is that, under what situations, the fixed field k(x1,…,xn)Gk(x1,…,xn)G is rational (= purely transcendental) over k  . This rationality problem has been studied by Hajja, Kang, Hoshi, Rikuna when n⩽3n⩽3. In this paper we will prove that k(x1,x2,x3,x4)Gk(x1,x2,x3,x4)G is rational over k   provided that the purely monomial action is decomposable. To prove this result, we introduce a new notion, the quasi-monomial action, which is a generalization of previous notions of multiplicative group actions. Moreover, we determine the rationality problem of purely quasi-monomial actions of K(x,y)GK(x,y)G over k   where k=KGk=KG.