1999/07/26 by Nicole Lemire, Lemire, Nicole, Martin Lorenz +1 · 1 citation
Computer Science · Mathematics · #13A50 #16G30 #16K40 #20C10 #20C30 #20J06 #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.AC #math.RA #msc:13A50 #msc:16G30 #msc:16K40 #msc:20C10 #msc:20C30 #msc:20J06
paper · pdf · doi:10.48550/arxiv.math/9907168
19 pages, AMS-LaTeX with XyPic
arxiv created 1999/07/26 · openalex publication_date 1999/07/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Sn denote the symmetric group on n letters. We consider the Sn-root lattice An-1 = (z1,...,zn) in Zn | z1+...+zn = 0, where Sn acts on Zn 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 Sn-invariant field of the exterior square of An-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 An-1 is equivalent to a permutation Sn-lattice. In the course of the proof of this result, we construct subgroups H of Sn, for all n that are not prime, so that the algebra of multiplicative H-invariants of An-1 has a non-trivial Picard group.