On certain lattices associated with generic division algebras

Abstract

Let S_n denote the symmetric group on n letters. We consider the S_n-root lattice A_{n-1} = {(z1,...,zn) in Z^n | z1+...+zn = 0}, where S_n acts on Z^n by permuting the coordinates, and its tensor, symmetric, and exterior squares. For odd values of n, we show that the tensor square is equivalent, in the sense of Colliot-Thelene and Sansuc, to the exterior square. Consequently, the rationality problem for generic division algebras, for odd values of n, amounts to proving stable rationality of the multiplicative S_n-invariant field of the exterior square of A_{n-1}. Furthermore, confirming a conjecture of Le Bruyn, we show that n=2 and n=3 are the only cases where the tensor square of A_{n-1} is equivalent to a permutation S_n-lattice. In the course of the proof of this result, we construct subgroups H of S_n, for all n that are not prime, so that the algebra of multiplicative H-invariants of A_{n-1} has a non-trivial Picard group.