Polarizations and Nullcone of Representations of Reductive Groups

The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f1, f2 . . . , fn be homogeneous invariant functions. Then the polarizations of f1, f2 . . . , fn define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). This means that there is a group homomorphism λ : C∗ → G such that limt→0 λ(t)x = 0 for all x ∈ L. This is then applied to many examples. A surprising result is about the group SL2 where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies. Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of SL2 on the n-fold tensor product C ⊗ C ⊗ · · · ⊗ C. Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n ≥ 3. (An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title “On the nullcone of representations of reductive group” is published in Pacific J. Math. 224 (2006), 119–140.) 1. Linear subspaces of the nullcone In this paper we study finite dimensional complex representations of a reductive algebraic group G. It is a well-known and classical fact that the nullcone NV of such a representation V plays a fundamental rôle in the geometry of the representation. Recall that NV is defined to be the union of all G-orbits in V containing the origin 0 in their closure. Equivalently, NV is the zero set of all non-constant homogeneous G-invariant functions on V . In a previous paper [KrW06] we have seen that certain linear subspaces of the nullcone play a central role for understanding its irreducible components. In this paper we will discuss arbitrary linear subspaces of the nullcone NV of a representation V of a reductive group G and show how they relate to questions about system of generators and systems of parameter for the invariants. We first recall the definition of a polarization of a regular function f ∈ O(V ). For k ≥ 1 and arbitrary parameters t1, . . . , tk we write (1) f(t1v1 + t2v2 + · · ·+ tkvk) = ∑ i1,i2,...,ik Pi1,··· ,ikf(v1, . . . , vk) · t i1 1 t i2 2 · · · t ik k . Date: October 15, 2007. The first author is partially supported by the Swiss National Science Foundation (Schweizerischer Nationalfonds), the second by an NSF summer grant.