Induction theorems on the stable rationality of the center of the ring of generic matrices

Following Procesi and Formanek, the center of the division ring of $n\times n$ generic matrices over the complex numbers $\mathbf C$ 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, $\mathbf C (G_n)$ and $\mathbf C(ZG\otimes _{ZN}G_n)$ 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 $\mathbf C$, in the cases $n=5$ and $n=7$.