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

Abstract

Let F be a field and let p be a prime. The problem we study is whether the center, C p , of the division ring of p × p generic matrices is stably rational over F . Given a finite group G and a Z G -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 Z S n -lattice, G n , such that C n is stably isomorphic to the fixed field under the action of S n of F ( G n ). Let H be a p -Sylow subgroup of S p . Let A be the root lattice, and let L=F( Z S p /H) . We show that there exists an action of S p on L( Z S P ⊗ Z H A) , twisted by an element α∈ Ext 1 S p ( Z S p ⊗ Z H A,L ∗ ) , such that L α ( Z S p ⊗ Z H A) S p is stably isomorphic to C p . The extension α corresponds to an element of the relative Brauer group of L over L H . Since Z S p ⊗ Z H A and Z S p /H are quasi-permutations, L( Z S p ⊗ Z H A) S p is stably rational over F . However, it is not known whether L α ( Z S p ⊗ Z H A) S p is stably rational over F . Thus the result represents a reduction on the problem since Z S p ⊗ Z H A is quasi-permutation; however, the twist introduces a new level of complexity.