On certain lattices associated with generic division algebras
| dc.creator | Lemire, Nicole | |
| dc.creator | Lorenz, Martin | |
| dc.date | 1999-07-26 | |
| dc.date.accessioned | 2026-07-07T05:30:04Z | |
| dc.date.available | 2026-07-07T05:30:04Z | |
| dc.description | 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. | |
| dc.description | 19 pages, AMS-LaTeX with XyPic | |
| dc.identifier | https://arxiv.org/abs/math/9907168 | |
| dc.identifier | http://arxiv.org/abs/math/9907168 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/78878 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Commutative Algebra | |
| dc.subject | 16G30; 20C10; 20C30; 13A50; 16K40; 20J06 | |
| dc.title | On certain lattices associated with generic division algebras | |
| dc.type | text |