Monomial actions of the symmetric group

Let F be a field, let G be a finite group and let M be a G-faithful ZG-lattice. We investigate the stable rationality of F(M)G over F, when G is Sp, the symmetric group on p letters, and p is a prime. It follows from work of Endo, Miyata, Lenstra, and Swan that G-faithful ZG-lattices are in the same flasque class if and only if they have G-isomorphic corresponding fields. Thus the study of flasque classes of ZSp-lattices plays a fundamental role in this investigation. Let N be the normalizer of a p-Sylow subgroup of Sp. We show that there are classes of ZSp-lattices for which induction restriction from N to Sp, does not affect the flasque class. In particular, we study the flasque class of a specific lattice, Gp, which has the property that F(Gp)Sp is stably isomorphic to the center of the division ring of generic matrices [Formanek, Linear and Multilinear Algebra 7 (1979) 203–212]. For a finite group G, lattices in the same genus are not in general in the same flasque class, however they are for G=Sn. We show that there is a class of ZSp-lattices, containing the genus of Gp, whose elements are in the flasque of Gp.