The Hilbert Null-cone on Tuples of Matrices and Bilinear Forms |
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Authors: | Matthias Bürgin Jan Draisma |
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Institution: | (1) Mathematical Institute, University of Basel, Rheinsprung 21, 4051 Basel, Switzerland;(2) Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands |
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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 (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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Keywords: | Representation theory Null-cone Matrices Bilinear forms |
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