Induction theorems on the stable rationality of the center of the
ring of generic matrices
Esther Beneish. 
Trans. Amer. Math. Soc. 350 (1998), 3571-3585.
MSC (1991): Primary 13A50, 20C10
                                               
Abstract: Following Procesi and Formanek, the center of the division ring of 
n x n generic matrices over the complex numbers C is stably equivalent to the
fixed field under the action of S_n, of the function field of the group 
algebra of a ZS_n-lattice, denoted by G_n. We study the question of the stable 
rationality of the center C_n over the complex numbers when n is a prime, 
in this module theoretic setting. Let N be the normalizer of an n-sylow 
subgroup of S_n. Let M be a ZS_n-lattice. We show that under certain 
conditions on M, induction-restriction from N to S_n does not affect the 
stable type of the corresponding field. In particular, C(G_n) and 
C(ZG \otimes_{ZN} G_n) are stably isomorphic and the isomorphism preserves 
the S_n-action. We further reduce the problem to the study of the 
localization of G_n at the prime n; all other primes behave well. We
also present new simple proofs for the stable rationality of C_n over C, in 
the cases n=5 and n=7.