Polarizations and Nullcone of Representations of Reductive Groups

Summary

We start with the following simple observation. Let V be a representation of a reductive group G, and let f ₁, f ₂, . . . , f _n be homogeneous invariant functions. Then the polarizations of f ₁, f ₂, . . . , f _n 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 $\lambda \mathbb{C} : * \to {\rm }G$ such that $\lim _{t \to 0} \lambda (t)x = 0$ 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 NumberingDepth="0" copies of SL₂ on the n-fold tensor product $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \cdot \cdot \cdot \otimes \mathbb{C}^2$. 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).

Mathematics Subject Classification (2000): 20G20, 20G05, 14L24