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

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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), 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 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).

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Correspondence to Matthias Bürgin.

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Bürgin, M., Draisma, J. The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms. Math. Z. 254, 785–809 (2006). https://doi.org/10.1007/s00209-006-0008-0

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