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[Vm]G for the diagonal action of G on Vm. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in K[Vm]G by the process of polarization and restitution from K[Vn]G. In particular, this means that if K[Vn]G is generated in degree ≤ d, then so is K[Vm]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 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 SLn × SLn on m-tuples of n × n matrices. In this case, we show that the invariants of degree ≤ n6 suffice to generate R(n, m) if the characteristic is larger than 2n6 + n2.