Polarizations and Nullcone of Representations of Reductive Groups

Abstract

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

SL2-modules of small homological dimension

“…Losik, Michor, and Popov [27], and independently Kraft and Wallach [26], showed for n > 1 that N (sV n ) is defined by the polarizations of any set of functions defining N (V n ).…”

Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

“…The nullcone plays a leading part in invariant theory. It is studied in [Ri5], [Po], [LMP], and recently in [KrW1] and [KrW2]. By [KrW1], V g is irreducible and has dimension 3(b g − rkg).…”

“…By [KrW1], V g is irreducible and has dimension 3(b g − rkg). Furthermore, N. Wallach and H. Kraft conjecture in [KrW2] that V g is an irreducible component of N g . Clearly, Theorem 1 confirms that conjecture.…”

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

“…This paper may be understood as an application of such approaches to the nullcone in the multi-vector representation of the symplectic group to obtain explicit combinatorial and representation theoretic descriptions of it. For the nullcone associated with representations of reductive groups and its interesting applications, we refer readers to [24,25].…”