On Certain Lattices Associated with Generic Division Algebras
Nicole Lemire and Martin Lorenz
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} = {(z_1, . . . ,z_n) \in Z^n  |  z_1+ . . .+ z_n = 0}, 
where  S_n acts on Z^n  by permuting the coordinates, and its tensor, 
symmetric, and exterior squares,  A_{n-1}^{\otimes 2}, Sym^2 A_{n-1} , and  
\Lambda^2 A_{n-1}. For odd values of  n , we show that   A_{n-1}^{\otimes 2}  
is equivalent to  \Lambda^2 A_{n-1}  in the sense of Colliot-Th\'el\`ene 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( \Lambda^2 A_{n-1})^{S_n} . Furthermore, 
confirming a conjecture of Le Bruyn, we show that n=2  and n=3  are the only 
cases where  A_{n-1}^{\otimes 2}  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.