Weyl's polarization theorem in positive characteristic

TL;DR

This paper investigates the validity of Weyl's polarization theorem in positive characteristic, demonstrating it holds in large enough characteristic for certain algebraic groups and providing bounds for invariants of matrix tuples.

Contribution

It extends Weyl's theorem to positive characteristic for reductive groups over integers, with explicit bounds for matrix invariants.

Findings

Weyl's theorem holds in large enough characteristic for good modules.

Explicit degree bounds are provided for generating invariants of matrix tuples.

Counterexamples exist in small characteristic, but the theorem is valid beyond certain thresholds.

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 $\leq d$, 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 $V^*$ 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 ${\rm SL}_n \times {\rm SL}_n$ on $m$-tuples of $n \times n$ matrices. In this case, we show that the invariants of degree $\leq n^6$ suffice to generate $R(n,m)$ if the characteristic is larger than $2n^6 + n^2$.