### Abstract

We describe the null-cone of the representation of G on M p , 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 S 2(V *) (symmetric bilinear forms), ¿2(V *) (skew bilinear forms), or (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M p is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M p . 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 M p 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).

Original language | English |
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Pages (from-to) | 785-809 |

Journal | Mathematische Zeitschrift |

Volume | 254 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2006 |

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## Cite this

Bürgin, M., & Draisma, J. (2006). The Hilbert null-cone on tuples of matrices and bilinear forms.

*Mathematische Zeitschrift*,*254*(4), 785-809. https://doi.org/10.1007/s00209-006-0008-0