MONOMIAL ACTIONS OF THE SYMMETRIC GROUP.

Esther Beneish
Department of Mathematics
Eastern Washington University
Cheney,  WA 99004

Abstract: 
Let F be a field, and let G be a finite group.  A rational extension of F on which G acts purely monomially, is of the form F(M) for some ZG-lattice M, where F(M) is the quotient field of the group algebra of the abelian group M. We investigate the stable rationality of the fixed field under the action of G of F(M) over F, when G is Sp, the Symmetric group on p letters, and p is a prime.  The study of flasque classes of ZSp-lattices plays a fundamental role in this investigation. Let N be the normalizer of a p-sylow 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.  We also present sufficient conditions for the flasque class of a ZSp-lattice to be zero, which implies that the corresponding fixed field is stably rational over F.  In particular we study the flasque class of a specific lattice, Gp, which has the property that the corresponding fixed field is stably isomorphic to the center of the generic division ring. 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 extend this to a larger class of ZSp-lattices containing the genus of Gp.

Status: Preprint.