The Hilbert null-cone on tuples of matrices and bilinear forms

M. Bürgin, J. Draisma

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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), ¿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 languageEnglish
Pages (from-to)785-809
JournalMathematische Zeitschrift
Volume254
Issue number4
DOIs
Publication statusPublished - 2006

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