Lattice invariants and the center of the generic division ring

Abstract:

Let $G$ be a finite group, let $M$ be a $ZG$-lattice, and let $F$ be a field of characteristic zero containing primitive $p$th roots of 1. Let $F(M)$ be the quotient field of the group algebra of the abelian group $M$. It is well known that if $M$ is quasi-permutation and $G$-faithful, then $F(M)^G$ is stably equivalent to $F(ZG)^G$. Let $C_n$ be the center of the division ring of $n\times n$ generic matrices over $F$. Let $S_n$ be the symmetric group on $n$ symbols. Let $p$ be a prime. We show that there exist a split group extension $G’$ of $S_p$ by a $p$-elementary group, a $G’$-faithful quasi-permutation $ZG’$-lattice $M$, and a one-cocycle $\alpha$ in $\operatorname {Ext}_{G’}^1(M,F^*)$ such that $C_p$ is stably isomorphic to $F_\alpha (M)^{G’}$. This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if $F$ is algebraically closed, there is a group extension $E$ of $S_p$ by an abelian $p$-group such that $C_p$ is stably equivalent to the invariants of the Noether setting $F(E)$.