Polarizations and nullcone of representations of reductive groups

Abstract

We start 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→0λ (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 ℂ2 ⊗ ℂ2 ⊗· · ·⊗ ℂ2. 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 groups” is published in Pacific J. Math. 224 (2006), 119–140).