Polarization of Separating Invariants

We prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue of Weyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups. Introduction We begin with a description of the standard invariant theory setting and recall the concepts of separating invariants and of polarization. Let K be any field and let V be a finite-dimensional vector space over K. We write K[V ] for the symmetric algebra of the dual space, V . If {x1, . . . , xk} is a basis of V , then K[V ] is the polynomial ring in the indeterminates x1, . . . , xk. Now suppose that G is any group acting linearly on V . Then there is a natural action of G on V ∗ which induces an action of G on K[V ]. The ring of invariants is the subring K[V ] of K[V ] consisting of those polynomials fixed pointwise by all of G: K[V ] := {f ∈ K[V ] | σ(f) = f for all σ ∈ G} . The main problem in invariant theory is to find a set of invariants S ⊂ K[V ] which generates K[V ] as a K-algebra. Such a set S is called a generating set. Since generating sets are often very complicated, and in some cases no finite generating sets exist, the concept of a separating set of K[V ] has emerged as a useful weakening of a generating set. Loosely speaking, a separating set is a set of invariants that has the same capabilities of separating G-orbits as all the Supported by the Swiss National Science Foundation. Partially supported by NSERC and ARP. 2000 Mathematics Subject Classification: 13A50, 14L24