Weyl's polarization theorem in positive characteristic

Abstract

Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m>0$, we study the invariant rings $K[V^m{]}^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in $K[V^m{]}^G$ by the process of polarization and restitution from $K[V^n{]}^G$. In particular, this means that if $K[V^n{]}^G$ is generated in degree $\le d$, then so is $K[V^m{]}^G$ no matter how large $m$ is. There are several explicit counterexamples to Weyl's theorem in positive characteristic. However, when $G$ is a (connected) reductive affine group scheme over $\mathbb{Z}$ and $V^{\ast }$ is a good $G$-module, we show that Weyl's theorem holds in sufficiently large characteristic. As a special case, we consider the ring of invariants $R(n,m)$ for the left-right action of ${\mathrm{SL}}_n\times {\mathrm{SL}}_n$ on $m$-tuples of $n\times n$ matrices. In this case, we show that the invariants of degree $\le n^6$ suffice to generate $R(n,m)$ if the characteristic is larger than $2n^6+n^2$.

17 pages