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No occurrence obstructions in geometric complexity theory

Authors: Peter Bürgisser, Christian Ikenmeyer and Greta Panova
Journal: J. Amer. Math. Soc. 32 (2019), 163-193
MSC (2010): Primary 68Q17, 05E10, 14L24
Published electronically: October 4, 2018
MathSciNet review: 3868002
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Abstract: The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes $\mathrm {VP}_{\mathrm {ws}}$ and $\mathrm {VNP}$. In 2001 Mulmuley and Sohoni suggested studying a strengthened version of this conjecture over complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of $\mathrm {GL}_{n^2}(\mathbb {C})$, which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni in 2001.

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Additional Information

Peter Bürgisser
Affiliation: Technische Universität Berlin, Berlin, Germany
MR Author ID: 316251

Christian Ikenmeyer
Affiliation: Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
MR Author ID: 911976

Greta Panova
Affiliation: University of Pennsylvania, Philadelphia, Pennsylvania; and Institute for Advanced Study, Princeton, New Jersey
MR Author ID: 964307

Keywords: Permanent versus determinant, geometric complexity theory, orbit closures, representations, plethysms, Young tableaux, tensors, highest weight vectors
Received by editor(s): March 9, 2017
Received by editor(s) in revised form: April 20, 2018
Published electronically: October 4, 2018
Additional Notes: The first author was partially supported by DFG grant BU 1371/3-2.
The third author was partially supported by NSF grant DMS-1500834
Article copyright: © Copyright 2018 American Mathematical Society