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On certain lattices associated with generic division algebras

2000; De Gruyter; Volume: 3; Issue: 4 Linguagem: Inglês

10.1515/jgth.2000.031

1435-4446

Nicole Lemire, Martin Lorenz,

Coding theory and cryptography

Let S n denote the symmetric group on n letters. We consider the S n-lattice Anˇ1a fOz1; ... ; znU A Z n j P i zia 0g, where S n acts on Z n by permuting the coordinates, and its squares A n2 nˇ1, Sym 2 A nˇ1, and 3 2 A nˇ1. For odd values of n, we show that A n2 nˇ1 is equivalent to 3 2 Anˇ1 in the sense of Colliot-Thelene and Sansuc (6). Consequently, the rationality prob- lem for generic division algebras amounts to proving stable rationality of the multipli- cative invariant field kO3 2 Anˇ1U S n (n odd). Furthermore, confirming a conjecture of Le Bruyn (16), we show that na 2 and na 3 are the only cases where A n2 nˇ1 is equivalent to a permuta- tion S n-lattice. In the course of the proof of this result, we construct subgroups HWS n, for all n that are not prime, so that the multiplicative invariant algebra kâAnˇ1a H has a non-trivial