The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms

Abstract

We describe the null-cone of the representation of G on Mp, where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S2(V*) (symmetric bilinear forms), Λ2(V*) (skew bilinear forms), or \(V^* \otimes V^*\) (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and Mp is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on Mp. Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in Mp is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).