The center of the generic division ring and twisted multiplicative group actions

AbstractLet F be a field and let p be a prime. The problem we study is whether the center, Cp, of the division ring of p×p generic matrices is stably rational over F. Given a finite group G and a ZG-lattice, we let F(M) be the quotient field of the group algebra of the abelian group M. Procesi and Formanek [Linear Multilinear Algebra 7 (1979) 203–212] have shown that for all n there is a ZSn-lattice, Gn, such that Cn is stably isomorphic to the fixed field under the action of Sn of F(Gn). Let H be a p-Sylow subgroup of Sp. Let A be the root lattice, and let L=F(ZSp/H). We show that there exists an action of Sp on L(ZSP⊗ZHA), twisted by an element α∈Ext1Sp(ZSp⊗ZHA,L∗), such that Lα(ZSp⊗ZHA)Sp is stably isomorphic to Cp. The extension α corresponds to an element of the relative Brauer group of L over LH. Since ZSp⊗ZHA and ZSp/H are quasi-permutations, L(ZSp⊗ZHA)Sp is stably rational over F. However, it is not known whether Lα(ZSp⊗ZHA)Sp is stably rational over F. Thus the result represents a reduction on the problem since ZSp⊗ZHA is quasi-permutation; however, the twist introduces a new level of complexity