Abstract. The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f 1 , f 2 . . . , fn be homogeneous invariant functions. Then the polarizations of f 1 , f 2 . . . , 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 :This is then applied to many examples. A surprising result is about the group SL 2 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 SL 2 on the n-fold tensor product C 2 ⊗ C 2 ⊗ · · · ⊗ 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 group" is published in Pacific J. Math. 224 (2006), 119-140.)
Linear subspaces of the nullconeIn 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 N V of such a representation V plays a fundamental rôle in the geometry of the representation. Recall that N V is defined to be the union of all G-orbits in V containing the origin 0 in their closure. Equivalently, N V 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 N V 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 t 1 , . . . , t k we write