On Certain Lattices Associated with Generic Division Algebras
(with Nicole Lemire)

Appeared in: J. Group Theory 3, 385-405 (2000).

Abstract:  Let  S_n  denote the symmetric group on  n  letters. We consider the S_n-lattice
A_n-1 = {(z₁, . . . ,z_n) e Zⁿ  |  z₁+ . . .+ z_n = 0}, where  S_n acts on Zⁿ  by permuting the coordinates,
and its tensor, symmetric, and exterior squares,  A_n-1^Ä2, Sym² A_n-1 , and  L² A_n-1.
For odd values of  n , we show that   A_n-1^Ä2  is equivalent to  L² A_n-1  in the sense of
Colliot-Thélène and Sansuc. Consequently, the rationality problem for generic division algebras,
for odd values of   n , amounts to proving stable rationality of the multiplicative invariant
field   k( L² A_n-1)^Sn . Furthermore, confirming a conjecture of Le Bruyn, we show that n=2  and
n=3  are the only cases where  A_n-1^Ä2  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 multiplicative invariant algebra  k[ A_n-1]^H   has a non-trivial Picard group.
 
 

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